Common Core Math

Students who feel difficult to solve the rotation problems can refer to this page and learn the techniques so easily. Rotation in Maths is turning an object in a circular motion on any origin or axis. Any object can be rotated in both directions ie., Clockwise and Anticlockwise directions. Generally, there are three rotation angles around the origin, 90 degrees, 180 degrees, and 270 degrees. One of the rotation angles ie., 270° rotates occasionally around the axis. Both 90° and 180° are the common rotation angles. Check out this article and completely gain knowledge about 180-degree rotation about the origin with solved examples.

180 Degree Rotation Around the Origin

When the point M (h, k) is rotating through 180°, about the origin in a Counterclockwise or clockwise direction, then it takes the new position of the point M’ (-h, -k). So, the 180-degree rotation about the origin in both directions is the same and we make both h and k negative.

Rule of 180° Rotation

• If the point (x,y) is rotating about the origin in 180-degrees clockwise direction, then the new position of the point becomes (-x,-y).
• If the point (x,y) is rotating about the origin in 180-degrees counterclockwise direction, then the new position of the point becomes (-x,-y).

Worked-Out Problems on 180-Degree Rotation About the Origin

Example 1:

Determine the vertices taken on rotating the points given below through 180° about the origin.

(i) P (6, 9)

(ii) Q (-5, 8)

(iii) R (-2, -6)

(iv) S (1, -3)

Solution:

The rule of 180-degree rotation is ‘when the point M (h, k) is rotating through 180°, about the origin in a Counterclockwise or clockwise direction, then it takes the new position of the point M’ (-h, -k)’. By applying this rule, here you get the new position of the above points:

(i) The new position of the point P (6, 9) will be P’ (-6, -9)

(ii) The new position of the point Q (-5, 8) will be Q’ (5, -8)

(iii) The new position of the point R (-2, -6) will be R’ (2, 6)

(iv) The new position of the point S (1, -3) will be S’ (-1, 3)

Example 2:

Put the point A (2, 3) on the graph paper and rotate it through 180° about the origin O. Calculate the new position of A’.

Solution:

Given coordinate is A = (2,3) after rotating the point towards 180 degrees about the origin then the new position of the point is A’ = (-2, -3) as shown in the above graph.

FAQs on 180 Degree Clockwise & Anticlockwise Rotation

1. What is the rule for 180° Rotation?

The rule for a rotation by 180° about the origin is (x,y)→(−x,−y).

2. Is turning 180 degrees clockwise different from turning 180 degrees counterclockwise?

Yes, both are different but the formula or rule for 180-degree rotation about the origin in both directions clockwise and anticlockwise is the same.

3. How the 180 degrees look like?

The measure of 180 degrees in an angle is known as Straight angles. Then the 180 degrees look like a Straight Line. 

An Irregular Figure is a figure that is not a standard geometric shape and you can’t calculate the area of them using the standard area formulas. However, some irregular shapes are formed using two or more standard geometric figures. Thus, to find the area of the irregular shapes we split them according to shapes whose formulas we know and then add the area of those figures.

Irregular Shapes Definition

Irregular shapes are polygons that have five or more sides of varying lengths. These shapes or figures can be decomposed further into squares, triangles, and quadrilaterals to evaluate the area.

How to Calculate Area of Irregular Figures?

There are various methods to calculate the Area of Irregular Shapes and we have outlined few popular ones in the below modules. They are as under

• Evaluating Area using Unit Squares
• Divide the Irregular Shapes into two or more regular shapes
• Divide the Irregular Shapes with Curves into two or more regular shapes

Evaluating Area using Unit Squares

Use this Technique if you are dealing with shapes that are curves apart from a perfect circle, semicircle, and are irregular quadrilaterals. In this technique, you need to divide the shape into unit squares. The total number of unit squares that fall within the shape determines the total area. Count the squares as 1 if the shaded region covers more than half to have an accurate estimation.

Divide the Irregular Shapes into two or more regular shapes

You can use this technique for irregular shapes that are a combination of known shapes such as triangles, polygons. Use the predefined formulas to find the area of such shapes and then add them up to know the total area.

Example:

The above figure has two regular shapes square, semi circle

We can find the areas of them individually and then team up to know the Area of Irregular Shape

Given Side of a Square = 4

Area of Square = S²

Substituting the side value in the area of square formula we get

Area of Square = 4²

= 16

Area of a Semi-Circle = \(\frac { 1 }{ 2 } \) π*r²

Diameter = 4

Radius = d/2 = 4/2 = 2

Area of a Semi-Circle = \(\frac { 1 }{ 2 } \)(3.14*2²)

= \(\frac { 1 }{ 2 } \)(3.14*4)

= \(\frac { 1 }{ 2 } \)12.56

= 6.28

Area of Irregular Shape = Area of Square + Area of Semi Circle

= 16+6.28

= 22.28

Divide the Irregular Shapes with Curves into two or more regular shapes

In this technique decompose the given irregular shape into multiple squares, triangles, or other quadrilaterals. Based on the Curve and Shapes, part of the figure can be a circle, semi-circle, or quadrant.

Given Irregular Shape can be divided into multiple squares, rectangles, semi-circle, etc.

We can split the above figure into two rectangles, half circle

Let us find the area of rectangle 1

Area of Rectangle 1 = Base * Height = 4 * 10 = 40

Later find the area of rectangle 2:

Area of Rectangle 2 = Base * Height = 3 * (8 – 4) = 12

Let us find the Area of Semi Circle

A = \(\frac { 1 }{ 2 } \) π*r²

Area of Semi Circle =  (1/2)(3.14)1² = 1.57

Sum up all the individual areas to get the Area of the Irregular Shape

Total Area = 40 + 12 + 1.57 = 53.57

Examples on Area of Irregular Shapes

1. Work out the Area for the following Shape?

Solution:

Given Irregular Shape can be further divided into two rectangles

Area of 1st rectangle = 10*5

= 50 cm²

Area of 2nd Rectangle = (9-5)*(10-6)

= 4*4

= 16 cm²

Therefore, the Area of Irregular Shape can be obtained by combining the areas of two rectangles

Area of Irregular Shape = Area of 1st rectangle + Area of 2nd Rectangle

= 50 cm² +16 cm²

= 66 cm²

2. Find the Area of Irregular Figure provided below?

Solution:

We can split the above Irregular Figure into known shapes like Rectangle, Squares.

Firstly, find the area of the rectangle = Length* Breadth

= 5*14

= 70 cm²

Area of Square = Side²

= 4²

= 16 cm²

Area of Irregular Figure = Area of Rectangle + Area of Square

= 70 cm²+ 16 cm²
= 86 cm²

One of the major roles played in the Probability concept in mathematics is a deck of 52 playing cards. The concept of Playing cards probability problems is solved on the basis of a well-shuffled pack of 52 cards. Whenever we face the probability topic in statistics, most of the problems with a well-shuffled pack of 52 playing cards. So, this article will make you learn what is the basic concept of cards, the formula, how to find the probability of playing cards, and worked-out problems on Playing Cards Probability.

Basic Stuff About Playing Cards Probability

In a deck or pack of playing cards, you will find the 52 playing cards which are divided into 4 suits of 13 cards. The shapes of those 4 suits are i.e. spades ♠ hearts ♥, diamonds ♦, clubs ♣. Also, these 4 suits are colored in two colors ie., red and black. Spades and clubs are black in color and the remaining Diamonds and hearts are Red in color.

The four different types of cards are shown in the picture given below.

How are 52 Cards Divided?

In each suit of playing cards includes an Ace, King, Queen, Jack or Knaves, 10, 9, 8, 7, 6, 5, 4, 3, and 2. In the pack of 52 cards, there are 12 face cards which are King, Queen, and Jack (or Knaves). Check out the below image and get full clarity about the 13 cards of each suit in the 52 playing cards.

Also, have a look at the below points to memorize easily about the pack of 52 playing cards:

1. Club – 13 cards
2. Heart – 13 cards
3. Spade – 13 cards
4. Diamond – 13 cards
5. No. of black cards – 26
6. No. of red cards – 26
7. No. of Ace cards (named as “A”) – 4
8. No. of Jack cards (named as “J” – 4
9. No. of Queen cards (named as “Q”) – 4
10. No. of King cards (named as “K”) – 4
11. No. of face cards (named as “J”, “Q” and “K”) – 12

Formula

Based on the classic definition of the probability the formula to find the probability with playing cards is as follows:

Probability = No. of favorable comes/ No.of all possible outcomes
          or
P(A) = n(A)/n(S)

In the case of 52 playing cards, n(S) = 52.

How to Find the Probability of Playing Cards?

In the following steps, we have explained how to find the probability of playing cards. So, follow the below steps to calculate the probability:

1. In the first step, you have to find the number of favorable events.
2. Next, identify the whole number of possible outcomes that can occur.
3. At last, divide the number of favorable events by the complete number of possible outcomes.

To help you in solving the playing cards probability, we have given some solved examples of probability below. Let’s get into the practice problems of playing cards probability.

Probability Cards Questions

Example 1:

A card is drawn at random from a well-shuffled deck of 52 cards. What is the probability that the drawn card is Queen?

Solution:

Assume E be the event of drawing a Queen Card.

In total there are 4 Queen cards in 52 playing cards

Then,

n(E) = 4

And, we know that

n(S) = 52

Now, apply the formula to find the playing cards probability for the drawn card,

P(E) = n(E) / n(S)

P(E) = 4/52 = 1/13

So, the probability of getting a Queen card is 1/13.

Example 2:

A card is drawn from a well-shuffled pack of 52 cards. Find the probability of getting a card of Heart.

Solution:

Let A represents the event of getting a Heart cart.

No.of Heart cards are 13

Therefore, n(A) = 13

Here, the total no. of possible outcomes of A is 52

P(A) = No.of favourable comes/ a total no. of possible outcomes of A

= n(A)/n(S)

=13/52

=1/4

Hence, the required probability of getting a card of Heart is 1/4.

In Mathematics there are four basic operations namely Addition, Subtraction, Multiplication, Division. Division Operation is one of the basic arithmetic operations. In Division Process there are two major parts namely Divisor and Dividend. Get to know the definition of each of the parts of the division explained in detail below. Know the Formula to find Dividend When Divisor, Quotient, and Remainer are given.

Dividend Divisor Quotient and Remainder Definitions

Dividend: The number or value that we divide is known as a dividend.

Divisor: The number that divides the dividend is known as a divisor

Quotient: The result obtained from the division process is known as a quotient

Remainder: The number left over after the division process is known as the remainder.

How to find the Dividend? | Dividend Formula

The Formula to Calculate Dividend is given as under

Dividend = Divisor x Quotient + Remainder

Whenever you divide a number with another number you will get the result.

a/b = c

Here “a” is the dividend and “b” is the divisor whereas “c” is the quotient. Thus, we can write it as follows

Dividend = Divisor*Quotient

Rearranging the Terms we can find the other terms if any two are known.

However, if you have a remainder after the division process then the formula is as follows

Dividend = Divisor*Quotient+Remainder

You can verify the same by applying the formula of dividend

We know Dividend = Divisor*Quotient+Remainder

487 = 32*15+7

487 = 480+7

487 = 487

So the answer is correct.

Check out Examples on Division of Integers to learn about the Terms Dividend, Divisor, Quotient, and Remainder in detail. Let us consider few examples to verify the answer of division.

Properties of Division

Below we have listed few division properties explained by considering few examples. Keep these below points in mind you will find the division problems much simple to solve.

Property 1: If you divide zero by a number the quotient is always zero.

Examples:

(i) 0 ÷ 2 = 0

(ii) 0 ÷ 10 = 0

(iii) 0 ÷ 15 = 0

(iv) 0 ÷ 214 = 0

(v) 0 ÷ 320 = 0

(vi) 0 ÷ 6132 = 0

Property 2: The division of a number by zero is not defined.

Example:

34 divided by 0 is not defined

Property 3: If we divide any number with 1 the quotient is always the number itself.

Examples

(i) 22 ÷ 1 = 22

(ii) 658 ÷ 1 = 658

(iii) 3250 ÷ 1 = 3250

(iv) 5483 ÷ 1 = 5483

Property 4: If we divide a non-zero number by itself the quotient is always one.

(i) 40 ÷ 40 = 1

(ii) 92 ÷ 92 = 1

(iii) 2137 ÷ 2137 = 1

(iv) 4130 ÷ 4130 = 1

You can refer to our Properties of Division of Integers to have an idea of more such properties.

Worried about How to Read and Write Large Numbers during your math calculations? Then you have come the right way where you will get complete details on Reading & Writing Large Numbers.  Know-How to Read and Write Large Numbers in Words, Numerals by referring to the later modules. Check out the Place Value Chart available below to know about different groups.

How to Read and Write Large Numbers?

Numbers are separated into groups or periods such as ones, tens, hundreds, thousands, millions, and so on. However, in each group, there exists three subgroups namely ones, tens, hundreds. Refer to the below examples provided so that you will have an idea of periods or groups. Thus, you can answer the questions on reading or writing large numbers into words, numerals, or vice versa.

Also, keep in mind that while reading or writing large numbers you need to begin from the left side with the largest group and then move towards the right side.

Reading and Writing Large Numbers in Hundreds and Thousands

(i) 4,017 – Four thousand seventeen

(ii) 6,129 -Six thousand twenty-nine

(iii) 9,780 – Nine thousand seven hundred eighty

(iv) 61,015 – Sixty-one thousand fifteen

(v) 82, 535 – Eighty two thousand five hundred thirty-five

(vi) 611,010 – Six hundred eleven thousand ten

(vii) 431,002 – Four hundred thirty one thousand two

Examples on Writing the Number Names in Words

(i) 528 – Five Hundred and Twenty-Eight

(ii) 3904 – Three Thousand Nine Hundred and Four

(iii) 49,103 – Forty-Nine Thousand One Hundred and Three

(iv) 713,084 – Seven Hundred Thirteen Thousand and Eighty-Four

(v) 3,183,012 –  Three million one hundred eighty-three thousand twelve

(vi) 75,868,501 – Seventy-five million eight hundred sixty-eight thousand five hundred and one

(vii) 427,215,640 – Four hundred twenty-seven million two hundred fifteen thousand six hundred forty

Writing the Numbers in Numerals Examples

(i) Eighty-four – 84

(ii) Five hundred twelve – 512

(iii) Seven thousand three – 7003

(iv) Fifty-two thousand three hundred and four – 52,304

(v) Seven hundred fifty-one thousand three hundred fifty-one – 751,351

(vi) Five million seven hundred five thousand five hundred eighty-three – 5,705,583

(vii) Nineteen million six hundred forty-nine thousand three hundred twenty-two – 19,649,322

Conversion Of Hours Into Seconds: Looking for a perfect guide to make you understand how to convert hrs to sec? then, this is the right page for you. Both Hours and seconds are units used to measure time. Here, we have explained completely about hours to seconds conversion with solved examples. Also, you can have a look at the Math Conversion Chart on our site to get a deeper idea of length, mass, capacity conversions along with the time conversions. Check the below modules and know what are the definitions for Hour and Second, simple conversion formula, and worked-out examples on how to convert hrs to sec.

Definitions for Hours & Seconds

Hour(hr): Hour is defined as a unit of time equal to 1/24 of a day or 60 minutes or 3,600 seconds. An hour is an SI unit of time for use with the metric system. The abbreviation of Hours is ‘hr’. For instance, 1 hour can be addressed as 1 hr. According to the conversion base 1 hour = 3600 seconds.

Seconds(sec):

First, the second was predicated on the length of the day but since 1967, the definition of a Seconds is exactly “the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom” (at a temperature of 0 K). Also, the second is a base unit of time. One second is equal to 0.00027777777777778 hr.

Formula to Hours to Seconds Conversion

The simple formula to convert hours into seconds is as follows:

Hours to Seconds Conversion Scale

In order to help you more understandable below, we have compiled a detailed explanation on how to convert Hrs to Sec easily. So, continue your read and learn the simple process of conversion of Hours into Seconds.

How to Convert Hours(hrs) to Seconds(sec)?

To Convert an Hours time measurement to a Seconds time measurement, multiply the time value by the conversion ratio and change the units to time. Let’s speak on the conversion of hr to sec even more deeply by following the lines and know how to convert hours to seconds easily.

As we discussed in the above definitions, One hour is equal to 3600 seconds, so the time in seconds is equal to the hours multiplied by 3,600.

Thus, the formula for Time conversion of Hr into Sec is Seconds = Hours × 3,600.

Once, you know the formula of conversion then substitute the given hours into the hrs to sec formula and calculate the hours to seconds conversion.

Seek help from the below-provided solved examples on Conversion of Hours into Seconds and understand the calculation process involved in converting from hr to sec.

Worked-out Examples on Hours into Seconds Conversion

Example 1:

Convert 3 hours into seconds?

Solution:

First, consider the given hours in the question ie., Hours = 3hrs

Now, apply the direct formula of hours to seconds conversion:

Seconds = Hours X 3600 sec

Seconds = 3 X 3600 sec

Seconds = 10,800 sec

Therefore, the conversion of hours into seconds for given 3hr = 10,800 sec.

Example 2:

Convert 4 hours 20 minutes into seconds.

Solution:

First, we have to convert 4hrs into seconds by multiply with the conversion ratio,

ie., Sec = hr x 3600 = 4 x 3600 = 14,400 sec

Next, we have to convert 20 minutes into seconds. As per the definition, 1 minute is equal to 60 seconds.

So, multiple by 60 minutes,

20 x 60 = 1200

Finally, to convert 4 hours 20 minutes into seconds, add them together:

ie., 14,400 + 1200 = 15600

Therefore, the conversion of 4 hours 20 minutes into seconds is 15600 sec. 

Common Conversions Facts About Hrs to Sec

Below are a few easy & quick Hours to Seconds Conversion Facts:

1 minute = 60 seconds
1 hour = 60 minutes
1 hour = 3600 seconds
12 hours = 720 minutes
12 hours = 43,200 seconds
1 day = 24 hours = 1440 minutes
1 day = 24 hours = 86,400 seconds

Hour to Second Conversion Table

Here is the conversion table of Hours(hrs) to Seconds(sec):

FAQs on Conversion Of Hours Into Seconds

1. What is the rule for converting hours to seconds?

The time in seconds is equal to the hours multiplied by 3600.

2. How many seconds has 1 hour?

As per the definition, one minute is equal to 60 seconds and 60 minutes in one hour implies that 1 hour = 60 * 60. Now, product it and you’ll get 1 hour has 3,600 seconds.

3. How do you convert Hr to Sec?

As there are 3600 seconds in an hour, to convert hours to seconds, multiply by 3600.

Wondering where to find Metric Conversions Chart and Customary Units Conversion of Length? Don’t worry as you will get all of them here. We know the Standard Unit of Length is Meter and is expressed in short as “m”. 1 m is usually divided into 100 equal parts and each part is named as Centimeter and written in short as “cm”.

Long distances are measured in Kilometers and 1 Km is equal to 1000 m and is written as “km”. Different Units of Length and their Equivalents are expressed in the further modules. Check out our Math Conversion Chart to learn about length, mass, capacity, conversions etc. In Most Cases, we use Kilometre (km), Metre (m), and Centimetre (cm) as units of length measurement. You can also check Customary Units of Length, its Chart in the below sections.

Length Conversion Charts

Different Units of Length Conversions are explained here which you can use as a part of your calculations. Make the most out of them whenever you need unit conversions.

Customary Units of Length are expressed here in the below table so that you can use the conversions of length.

Solved Examples on Unit Conversions

1. Convert 0.7 m to cm?

Solution:

We know 1 m = 100 cm

0.7m = 0.7*100

= 70 Cm

Therfore, 0.7m converted to cm is 70 cm.

2. Convert 5 m 16 cm to m?

Solution:

We know 1 cm = \(\frac { 1 }{ 100 } \) m

5m 16 cm = 5m +\(\frac { 16 }{ 100 } \)m

= 5m+0.16m

= 5.16m

Therefore, 5m 16 cm converted to m is 5.16m

3. Convert 14 km 350 m into km?

Solution:

We know 1 km = 1000m

1m = \(\frac { 1 }{ 1000 } \) km

14km 350m = 14 km+\(\frac { 350 }{ 1000 } \) km

= 14km+0.35km

= 14.35 km

14 km 350 m converted to m is 14.35km

4. The length of a square tile is 120 cm. What will be the length of the tile strip in millimeters if 12 tiles are kept in a line?

Solution:

Length of Square Tile = 120 cm

Length of 12 Tile Strips in mm = 120*12

= 1440 mm

Therefore, the Length of the Tile Strip in mm is 1440mm.

Three-Dimensional Figures are shapes that consist of 3 dimensions such as length, breadth, and height. Three-Dimensional shapes also called solids. The length, breadth, and height are the three important measurements of 3-dimensional figures. There are different 3-dimensional shapes used in real-time. They are cuboids, cubes, Prisms, Pyramids, cylinders and cones, etc.

Solid Shapes

Solid shapes are the fixed objects they have fixed size, shape, and space. Let us check different examples of Solid Shapes to deeply understand the Solid Geometrical Figures.

Surface Area and Volume of Three-Dimensional shapes

Surface Area is defined as the complete area of the surface of the three-dimensional object. It is measured in square units. The surface area can be calculated using three different classifications.

• Curved Surface Area (CSA) is the area present in all the curved regions.
• Lateral Surface Area (LSA) is the area of all the flat surfaces and all the curved regions excluding base areas.
• Total Surface Area (TSA) is the area of all the surfaces including the base of a Three-Dimensional object.

The volume of the 3D shape is explained as the total space occupied by the three-dimensional object. It is measured in terms of cubic units and denoted by V.

Faces, Vertices, and Edges of a 3-Dimensional Shape

Have a look at the Faces, Vertices, and Edges of a 3-dimensional object.

• A solid consists of a flat part on it. Each flat part of a solid is known as the Face of a solid.
• The corner or Vertex is an end where three faces of a solid join together. Vertices are the plural form of the vertex.
• When two faces of a solid meet in a line called an Edge.

Types of Three Dimensional Shapes(3D Shapes)

Here we are going to discuss the list of three-dimensional shapes, their properties, and formulas. We even took examples for a better understanding of the concept.

1. Cuboid

A cuboid is also known as a rectangular prism consists of rectangle faces. The cuboid has 90 degrees angles each. Also, it has 8 vertices, 12 edges, 6 faces.
The formula of surface area and volume of a cuboid is given below.
Surface Area of a Cuboid = 2(lb + bh + lh) Square units
The volume of a Cuboid = lbh Cubic units
Examples of Cuboid are a box, a book, a matchbox, a brick, a tile, etc.,

Example:
Let us consider the below figure to completely understand a Cuboid

(i) Faces of a Cuboid: A cuboid consists of 6 faces. From the given figure, the 6 faces of the cuboid are PQRS, EFGH, PSHE, QRGF, PQFE, and SRGH.
(ii) Vertices of a Cuboid: A cuboid has 8 vertices. From the given figure, the 8 vertices of the cuboid are P, Q, R, S, E, F, G, H.
(iii) Edges of a Cuboid: A cuboid has 12 edges. From the given figure, the 12 edges of the cuboid are PQ, QR, RS, SP, EF, FG, GH, HE, PE, SH, QF, RG.

2. Cube

A Cube is of solid shape and consists of 6 square faces. The Cube all edges are equal. Also, it has 8 vertices, 12 edges, 6 faces.
The formula of surface area and volume of a Cube is given below.
Surface Area of a Cube = 6a² Square units
The volume of a Cube = a³ Cubic units

Example:
Let us consider the below figure to completely understand a Cube.

(i) Faces of a Cube: A cube consists of 6 faces. From the given figure, the 6 faces of the cube are PQRS, EFGH, PSHE, QRGF, PQFE, and SRGH.
(ii) Vertices of a Cube: A cube consists of 8 vertices. From the given figure, the 8 vertices of the cube are P, Q, R, S, E, F, G, H.
(iii) Edges of a Cube: A cube consists of 12 edges. From the given figure, the 12 edges of the cube are PQ, QR, RS, SP, EF, FG, GH, HE, PE, SH, QF, RG.

3. Prism

A prism has two equal ends, flat faces or surfaces, and also it has an identical cross-section across its length. If the cross-section of a prism looks like a triangle, then the prism is called a triangular prism. The prism will not have any curve. Also, it has 6 vertices, 9 edges, 5 faces (2 triangles and 3 rectangles).
The formula of surface area and volume of a Prism is given below.
Surface Area of a prism = 2(Base Area) + (Base perimeter × length) square units
The volume of a prism = Base Area × Height Cubic units

Example:
Let us consider the below figure to completely understand a triangular prism.

(i) Faces of a Triangular Prism: A triangular prism consists of 2 triangular faces and 3 rectangular faces. From the given figure, 2 triangular faces are ∆PQR and ∆STV, 3 rectangular faces are PQTS, PSVR, and RSTV.
(ii) Vertices of a Triangular Prism: A triangular prism consists of 6 vertices. From the given figure, the 6 vertices of the triangular prism are P, Q, R, S, T, V.
(iii) Edges of a Triangular Prism: A triangular prism consists of 9 edges. From the given figure, the 9 edges of the triangular prism are PQ, QR, RP, ST, TV, VS, PS, QT, RV.

4. Pyramid

A pyramid has a triangular face on the outside and its base is square, triangular, quadrilateral, or in the shape of any polygon. Also, it has 5 vertices, 8 edges, 5 faces.
The formula of surface area and volume of a Prism is given below.
Surface Area of a Pyramid = (Base area) + (1/2) × (Perimeter) × (Slant height) square units
The volume of a Pyramid = 1/ 3 × (Base Area) × height Cubic units

Example:
1. Let us consider the below figure to completely understand a Square Pyramid.

(i) Vertices of a Square Pyramid: A square pyramid consists of 5 vertices. From the given figure, OPQRS is a square pyramid having O, P, Q, R, S as its vertices.
(ii) Faces of a Square Pyramid: A square pyramid consists of faces one of which is a square face and the rest four are triangular faces. From the given figure, OPQRS is a square pyramid having PQRS as its square face and OPS, ORS, OQR, and OPQ as its triangular faces.
(iii) Edges of a Square Pyramid: A square pyramid consists of 8 edges. From the given figure, the square pyramid OPQRS has 8 edges, namely, PQ, QR, RS, SP, OP, OQ, OR, and OS.

2. Let us consider the below figure to completely understand a Rectangular Pyramid.

(i) Vertices of a Rectangular Pyramid: A Rectangular pyramid consists of 5 vertices. From the given figure, OPQRS is a Rectangular pyramid having O, P, Q, R, S as its vertices.
(ii) Faces of a Rectangular Pyramid: A Rectangular pyramid consists of 1 rectangular face and 4 triangular faces. From the given figure, OPQRS is a rectangular pyramid having PQRS as its rectangular face and OPS, OSR, OPQ, OQR as its triangular faces.
(iii) Edges of a Rectangular Pyramid: A rectangular pyramid consists of 8 edges. From the given figure, the rectangular pyramid OPQRS has 8 edges, namely, PQ, QR, RS, SP, OP, OQ, OR, and OS.

3. Let us consider the below figure to completely understand a triangular Pyramid.

(i) Vertices of a triangular Pyramid: A triangular pyramid consists of 4 vertices. From the given figure, PQRS is a Rectangular pyramid having P, Q, R, S as its vertices.
(ii) Faces of a triangular Pyramid: A triangular pyramid consists of 4 triangular faces. From the given figure, PQRS is a rectangular pyramid having PQR, OPQ, OPR, and OQR triangular faces.
(iii) Edges of a triangular Pyramid: A triangular pyramid consists of 6 edges. From the given figure, the triangular pyramid PQRS has 6 edges, namely, OP, OQ, OR, PQ, PR, QR.

Cylinder

A cylinder is explained as a figure that has two circular bases connected by a curved surface. The cylinder will not consist of any vertices. Also, it has 1 curved face, 2 edges, 2 flat faces.
The formula of surface area and volume of a cylinder is given below.
Surface Area of a cylinder = 2πr(h +r) Square units
The curved surface of a cylinder = 2πrh Square units
The volume of a Cylinder = πr2 h Cubic units

Cone

A cone is defined as a three-dimensional figure, which has a circular base and has a single vertex. The cone decreases smoothly from the circular flat base to the top point. Also, it has 1 vertex, 1 edge, 1 flat face – circle, 1 curved face.
The formula of surface area and volume of a cylinder is given below.
Surface Area of a cone = πr(r +√(r²+h²)  Square units
The curved surface of the area of a cone = πrl Square units
Slant height of a cone = √(r²+h²) Cubic units
The volume of a cone = ⅓ πr²h Cubic units

Sphere

A sphere appears round in shapes and every point on its surface is equidistant from the center point. The distance from the center to any point of the sphere is called the radius of the sphere. Also, it has No vertex, No edges, 1 curved face.
The formula of surface area and volume of a cylinder is given below.
The Curved Surface Area of a Sphere = 2πr² Square units
The Total Surface Area of a Sphere = 4πr² Square units
The volume of a Sphere = 4/3(πr³) cubic units

Types of Fractions and their rules, methods, and formulae are defined here. Know the various types of fractions along with their usage in various situations. Refer to the terminology involved in fractions and also know the problems involved in it. Follow fraction rules and real-life scenarios of fractions. Check the below sections to find examples, rules, methods, etc.

Types of Fractions | What are Fractions?

Before going to know about types of fractions, first know what are fractions and how they work in real life. Fractions are derived from the Latin word “fractus” which means the number or quantity that represents the part or portion of the whole. In the language of layman, a fraction means a number that describes the size of the parts of a whole. Fractions are generally declared with the numerator displaying above the line and below the line, the denominator will be displayed. The terms numerators and denominators are also used in other fractions like mixed, complex, and compound.

Fractions can be written as the equal or same number of parts being counted which is called the numerator over the number or quantity of parts in whole which is called the denominator. There are three major types of fractions which are proper fractions, improper fractions, and mixed fractions. These fractions are divided based on numerator and denominator. Apart from these major fractions, there are also other fractions such as like fractions, unlike fractions, equivalent fractions, etc.

Proper, Improper, and Mixed fractions are defined as single fractions and the remaining fractions determine the comparison of fractions.

Fraction Definition and Terminology

The fraction is considered as the ratio between two numbers. Fractions are defined by a/b. a is called the numerator which means the equal number of parts that are counted. b is called the denominator which means a number of parts in the whole. The numerator and denominator are divided with a line. The line denoted the separation between the numerator and the denominator.

Fraction Types

There are various types of fractions available. We have listed a few of them and explained their definitions, examples in detail. They are as such

1. Proper Fraction:

A proper fraction is that where the value of the numerator is less than the value of the denominator

If you include a numerator, a denominator and a line in between, then it is called a fraction. Proper Fraction is defined as Numerator < Denominator. The value of the proper fraction is always less than 1.

Example:

1/2, 9/15,30/45 are the proper fractions

2. Improper Fraction:

An improper fraction is that where the value of the denominator is less than the value of the numerator. Improper fractions are defined as Numerator > Denominator. Each natural number can be written in fractions in which the denominator is always 1. For example: 20/1,40/1,35/1. The value of the improper fraction is always greater than 1.

Examples: 

3/2, 16/10,45/15 are the improper fractions

3. Mixed Fraction:

A mixed fraction is the combination of a natural number and a fraction. These fractions are improper fractions. Mixed fractions can easily be converted into improper fractions and also mixed fractions can also be converted to improper fractions. The mixed fraction is always greater than 1.

Examples:

3 4/3, 4 5/4, 6 2/3

4. Like Fraction:

If the fractions have the same denominator, then they are called like fractions. For additional simplifications, we can easily make with the like fractions. Addition, Subtraction, Division and multiplication operations can be made easily on like factors.

Examples:

1/2,3/2.5/2,7/2,9/2 are like fractions.

5. Unlike Fractions:

If the factors have different or unique denominators, then they are called, unlike fractions. Simplification of fractions is a lengthy process, therefore we factorize the denominators and then simplify the numerators.

Suppose that we have to add two fractions 1/2 and 1/3.

As the denominators are different, take the LCM of 2 and 3 is equal to 6.

Now, we multiply 1/2 and 1/3 by 2. Multiply it both in numerator and denominator.

Therefore, the fraction becomes 3/6 and 2/6

Now if add 3/6 and 2/6, we get 5/6

Examples:

1/3,1/5,1/7 are unlike fractions

6. Equivalent Fractions:

When more fractions have a similar or same result even after simplification, they represent a similar portion of the whole. Those fractions are equal or similar to each other which are called equivalent fractions.

Examples:

1/2 and 2/4 are equivalent to each other.

1/3 and 3/9 are also equivalent to each other.

How to Convert Improper Fractions into Mixed Fractions?

To convert an improper fraction into mixed fractions, the numerator is divided by the denominator, and the quotient is written as the whole number and the remainder as the numerator.

Example:

Convert the fraction 17/4 into a mixed fraction?

Solution:

To solve the above problem, the steps undertaken are

• Divide the numerator of the given fraction 17 by the denominator of the fraction 4.
• After solving, the quotient is 4 and the remainder is 1.
• Now, combine the whole number 4 with the fraction of 1/4.
• Therefore, the mixed fraction is 4 1/4.

How to Convert Mixed Fraction to Proper Fraction?

The mixed fraction is converted to a proper fraction by multiplying the denominator of the fraction with the whole number and the product is added to the numerator.

The steps that are to be followed to convert mixed fraction as a proper fraction are:

• Multiply the denominator with the whole number.
• Suppose that 2 1/3 is an improper fraction.
• In the above equation, 2 is the whole number and 3 is the denominator.
• 2 * 3 = 6
• Add the product of the numerator
• 6 + 1 = 7
• Now, after adding the product, numerator changes to 7 and denominator changes to 3.
• Now, write the result as an improper fraction as 7/3

How to add Unlike Fractions?

To add the unlike fractions, first, we have to convert them to like fractions. The steps that are involved in adding the unlike fractions are:

1. First, calculate the LCM of both the denominators.
2. The result of LCM will be the denominator of fractions.
3.  Now, we have to calculate the equivalent value of the 1st fraction. To calculate the equivalent, first, divide the LCM calculated in the previous step with the denominator of the 1st fraction. Now, multiply the numerator with the denominator value.
4. In the same way, calculate the equivalent value of the second fraction. To find the equivalent value of the 2nd fraction, divide that LCM that is calculated in the first step by the denominator of the second fraction. Now multiply it with the numerator, therefore both the fractions have the same numerator.
5. Finally, add both the values of the numerator as shown in the previous section.

The Perimeter and Area of Rectangle are two important formulas in Mensuration. It calculates the space occupied by the rectangle and the length of boundaries of the rectangle. In this article, students can come across the concept of area and perimeter of rectangle deeply.

A rectangle is a quadrilateral with two equal sides and two parallel lines and four right angles. The concept of area and perimeter of rectangle formulas are explained with examples. Some of the examples of different shapes are given below.

What is Perimeter and Area of a Rectangle?

Perimeter: Perimeter of the rectangle is the sum of all the sides of the rectangle. The rectangle has two lengths and two breadths. To find the perimeter of the rectangle we have to add the length and breadth. It is measured in units. It is denoted by P.

Area: The area of the rectangle formula helps to calculate the length and breadth of the two-dimensional closed figure. To find the area of the rectangle we have to multiply the length and breadth of the rectangle. It is measured in square units. It is denoted by A.

Properties of Rectangle

• A Rectangle has two equal sides
• The rectangle is a quadrilateral
• The diagonals of the rectangle have the same length.
• The diagonal of the rectangle bisect each other.
• Sum of all the four angles is 360º
• Each angle of the rectangle is 90º
• If the sides of the rectangle are l and b then the diagonal of the rectangle d = √l² + b²
• The opposite sides of the rectangle are parallel.

Derivation of Perimeter of a Rectangle

The perimeter of the rectangle is the sum of all four sides.
P = l + l + b + b
P = 2l + 2b
P = 2(l + b)
Thus the perimeter of the rectangle is = 2(l + b)

Area of the Rectangle

The area of the rectangle is the product of length and breadth.
A = l × b

Perimeter and Area of Rectangle Formula

• Area of Rectangle = l × b
• Perimeter of Rectangle = 2(l + b)
• Length of the rectangle = A/b
• The breadth of the Rectangle = A/l
• Diagonal of the Rectangle = √l² + b²

Solved Problems on Area and Perimeter of a Rectangle

The formula of Perimeter and Area of Rectangle is explained step by step here with examples. Go through the below questions and solve the problems using the Area and Perimeter of the Rectangle formula.

1. Find the length of the rectangular plot whose breadth is 11 cm and the area is 165 cm². Also, find the perimeter of the rectangle?

Solution:

Given,
Breadth = 11 cm
Area = 165 cm²
Area of the rectangle = l × b
165 sq. cm = l × 11 cm
l = 165/11
l = 15 cm
Thus the length of the rectangle is 15 cm.
We know that,
Perimeter of the rectangle = 2(l + b)
P = 2(15 + 11)
P = 2(26)
P = 52 cm
Thus the perimeter of the rectangle is 52 cm.

2. Find the Perimeter of the Rectangle whose length is 10 cm and breadth is 8 cm?

Solution:

Given, Length = 10 cm
Breadth = 8 cm
We know that,
Perimeter of the rectangle = 2(l + b)
P = 2(10 cm + 8 cm)
P = 2(18 cm)
P = 36 cm
Therefore the perimeter of the rectangle is 36 cm.

3. Find the area of the rectangle whose length is 14 meter and width is 10 meters?

Solution:

Given,
Length = 14 meter
Width = 10 meter
We know that,
Area of the rectangle = l × w
A = 14 m × 10 m
A = 140 sq. meters
Therefore the area of the rectangle is 140 square meters.

4. A rectangular plot has its length of 16 cm and a perimeter of 60 cm. Find the width of the rectangular plot?

Solution:

Given,
Length = 16 cm
Perimeter = 60 cm
Width = ?
We know that
The perimeter of the rectangle = 2(l + w)
60 cm = 2(16 cm + w)
16 + w = 60/2
16 + w = 30
w = 30 – 16
w = 14 cm
Thus the width of the rectangular plot is 14 cm.

5. Find the area and perimeter of the rectangle whose length and breadth are 12 m and 6 m?

Solution:
Given,
length = 12 m
breadth = 6 m
We know that,
Area of the rectangle = l × b
A = 12 m × 6 m
A = 72 sq. meters
Now find the perimeter of the rectangle
P = l + l + b + b
P = 12 m + 12 m + 6 m + 6 m
P = 24 m + 12 m
P = 36 m
Therefore the area and perimeter of the rectangle is 72 square meters and 36 meters.

FAQs on Area and Perimeter of Rectangle

1. How to find the perimeter of a rectangle?

The perimeter of the rectangle can be calculated by adding all the sides of the rectangle.

2. What is the area of the rectangle?

The area of the rectangle is defined as the space occupied by the rectangle or closed figure. It is the product of length and breadth.

3. What is the unit for the perimeter of a rectangle?

The unit for the perimeter of the rectangle is cm or meters.