Area - Examples, Exercises and Solutions

Question Types:
Area of a Circle: A shape consisting of several shapes (requiring the same formula)Area of a Circle: Calculate The Missing Side based on the formulaArea of a Deltoid: Calculation using percentagesArea of a Deltoid: Subtraction or addition to a larger shapeArea of a Deltoid: Verifying whether or not the formula is applicableArea of a Parallelogram: Using congruence and similarityArea of a Parallelogram: Using external heightArea of a Rectangle: A shape consisting of several shapes (requiring the same formula)Area of a Rectangle: Calculation using the diagonalArea of a Rectangle: Extended distributive lawArea of a Rhombus: Extended distributive lawArea of a Rhombus: Verifying whether or not the formula is applicableArea of a Trapezoid: Finding Area based off Perimeter and Vice VersaArea of a Triangle: Ascertaining whether or not there are errors in the dataArea of a Triangle: Calculating in two waysArea of a Triangle: Using congruence and similarityArea of a Triangle: Worded problemsArea of a Circle: Increasing a specific element by addition of.....or multiplication by.......Area of a Rectangle: Subtraction or addition to a larger shapeArea of a Rectangle: Using Pythagoras' theoremArea of a Rhombus: Calculation using percentagesArea of a Rhombus: Finding Area based off Perimeter and Vice VersaArea of a Rhombus: Using Pythagoras' theoremArea of a Rhombus: Using variablesArea of a Trapezoid: Subtraction or addition to a larger shapeArea of a Trapezoid: Suggesting options for terms when the formula result is knownArea of a Circle: Using Pythagoras' theoremArea of a Rectangle: Using short multiplication formulasArea of a Rhombus: Using ratios for calculationArea of the Square: Increasing a specific element by addition of.....or multiplication by.......Area of a Circle: Calculating parts of the circleArea of a Circle: Using additional geometric shapesArea of a Deltoid: Using Pythagoras' theoremArea of a Parallelogram: Using Pythagoras' theoremArea of a Triangle: Subtraction or addition to a larger shapeArea of a Circle: Finding Area based off Perimeter and Vice VersaArea of a Circle: Subtraction or addition to a larger shapeArea of a Parallelogram: Calculating in two waysArea of a Rectangle: Using ratios for calculationArea of a Triangle: How many times does the shape fit inside of another shape?Area of the Square: Worded problemsArea of a Circle: Applying the formulaArea of a Parallelogram: Verifying whether or not the formula is applicableArea of a Trapezoid: Using Pythagoras' theoremArea of a Triangle: Using Pythagoras' theoremArea of a Triangle: Using ratios for calculationArea of a Parallelogram: Using ratios for calculationArea of a Rectangle: Worded problemsArea of a Trapezoid: Using ratios for calculationArea of a Deltoid: Using external heightArea of a Trapezoid: Using variablesArea of a Rhombus: Applying the formulaArea of a Rhombus: Calculate The Missing Side based on the formulaArea of a Triangle: Using additional geometric shapesArea of a Deltoid: Using ratios for calculationArea of a Rectangle: Calculate The Missing Side based on the formulaArea of a Rectangle: Using additional geometric shapesArea of a Triangle: Using variablesArea of a Deltoid: Identifying and defining elementsArea of a Parallelogram: Using additional geometric shapesArea of a Trapezoid: Using additional geometric shapesArea of a Deltoid: Using variablesArea of a Parallelogram: Using variablesArea of a Rectangle: Applying the formulaArea of the Square: True / falseArea of a Deltoid: Using additional geometric shapesArea of a Parallelogram: Finding Area based off Perimeter and Vice VersaArea of the Square: Express usingArea of a Triangle: Finding Area based off Perimeter and Vice VersaArea of the Square: Applying the formulaArea of a Deltoid: Applying the formulaArea of a Parallelogram: Calculate The Missing Side based on the formulaArea of a Rectangle: Finding Area based off Perimeter and Vice VersaArea of a Deltoid: Finding Area based off Perimeter and Vice VersaArea of a Triangle: Calculate The Missing Side based on the formulaArea of a Deltoid: Calculate The Missing Side based on the formulaArea of the Square: Calculate The Missing Side based on the formulaArea of a Trapezoid: Calculate The Missing Side based on the formulaArea of a Triangle: Applying the formulaArea of a Parallelogram: Applying the formulaArea of a Rectangle: Using variablesArea of a Trapezoid: Applying the formula

In this article, we will learn what area is, and understand how it is calculated for each shape, in the most practical and simple way there is.
Shall we start?

What is the area?

Area is the definition of the size of something. In mathematics, which is precisely what interests us now, it refers to the size of some figure.
In everyday life, you have surely heard about area in relation to the surface of an apartment, plot of land, etc.
In fact, when they ask what the surface area of your apartment is, they are asking about its size and, instead of answering with words like "big" or "small" we can calculate its area and express it with units of measure. In this way, we can compare different sizes.

Units of measurement of area

Large areas such as apartments are usually measured in meters, therefore, the unit of measurement will be m2 m^2 square meter.
On the other hand, smaller figures are generally measured in centimeters, that is, the unit of measurement for the area will be cm2 cm^2 square centimeter.
Remember:
Units of measurement for the area in cm=>cm2 cm => cm^2
Units of measurement for the area m=>m2 m=>m^2

Detailed explanation

Suggested Topics to Practice in Advance

  1. Square

Practice Area

Question 1

Calculate the area of the parallelogram according to the data in the diagram.

101010777AAABBBCCCDDDEEE

Question 2

The width of a rectangle is equal to 15 cm and its length is 3 cm.

Calculate the area of the rectangle.

Question 3

Calculate the area of the trapezoid.

555141414666

Question 4

Calculate the area of the triangle below, if possible.

8.58.58.5777

Question 5

Look at the rectangle ABCD below.

Side AB is 4.5 cm long and side BC is 2 cm long.

What is the area of the rectangle?
4.54.54.5222AAABBBCCCDDD

Examples with solutions for Area

Exercise #1

Calculate the area of the parallelogram according to the data in the diagram.

101010777AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

Since we know that ABCD is a parallelogram, according to the properties of parallelograms, each pair of opposite sides are equal and parallel.

Therefore CD=AB=10 CD=AB=10

We will calculate the area of the parallelogram using the formula of side multiplied by the height drawn from that side, so the area of the parallelogram is equal to:

SABCD=10×7=70cm2 S_{ABCD}=10\times7=70cm^2

Answer

70

Exercise #2

The width of a rectangle is equal to 15 cm and its length is 3 cm.

Calculate the area of the rectangle.

Video Solution

Step-by-Step Solution

To calculate the area of the rectangle, we multiply the length by the width:

15×3=45 15\times3=45

Answer

45

Exercise #3

Calculate the area of the trapezoid.

555141414666

Video Solution

Step-by-Step Solution

We use the formula (base+base) multiplied by the height and divided by 2.

Note that we are only provided with one base and it is not possible to determine the size of the other base.

Therefore, the area cannot be calculated.

Answer

Cannot be calculated.

Exercise #4

Calculate the area of the triangle below, if possible.

8.58.58.5777

Video Solution

Step-by-Step Solution

The formula to calculate the area of a triangle is:

(side * height corresponding to the side) / 2

Note that in the triangle provided to us, we have the length of the side but not the height.

That is, we do not have enough data to perform the calculation.

Answer

Cannot be calculated

Exercise #5

Look at the rectangle ABCD below.

Side AB is 4.5 cm long and side BC is 2 cm long.

What is the area of the rectangle?
4.54.54.5222AAABBBCCCDDD

Video Solution

Step-by-Step Solution

We begin by multiplying side AB by side BC

We then substitute the given data and we obtain the following:

4.5×2=9 4.5\times2=9

Hence the area of rectangle ABCD equals 9

Answer

9 cm²

Question 1

Look at rectangle ABCD below.

Side AB is 10 cm long and side BC is 2.5 cm long.

What is the area of the rectangle?
1010102.52.52.5AAABBBCCCDDD

Question 2

Given the following rectangle:

666999AAABBBDDDCCC

Find the area of the rectangle.

Question 3

Given the following rectangle:

888444AAABBBDDDCCC

Find the area of the rectangle.

Question 4

Given the following rectangle:

222555AAABBBDDDCCC

Find the area of the rectangle.

Question 5

Given the following rectangle:

111111777AAABBBDDDCCC

Find the area of the rectangle.

Exercise #6

Look at rectangle ABCD below.

Side AB is 10 cm long and side BC is 2.5 cm long.

What is the area of the rectangle?
1010102.52.52.5AAABBBCCCDDD

Video Solution

Step-by-Step Solution

Let's begin by multiplying side AB by side BC

If we insert the known data into the above equation we should obtain the following:

10×2.5=25 10\times2.5=25

Thus the area of rectangle ABCD equals 25.

Answer

25 cm²

Exercise #7

Given the following rectangle:

666999AAABBBDDDCCC

Find the area of the rectangle.

Video Solution

Step-by-Step Solution

We will use the formula to calculate the area of a rectangle: length times width

9×6=54 9\times6=54

Answer

54

Exercise #8

Given the following rectangle:

888444AAABBBDDDCCC

Find the area of the rectangle.

Video Solution

Step-by-Step Solution

Let's calculate the area of the rectangle by multiplying the length by the width:

4×8=32 4\times8=32

Answer

32

Exercise #9

Given the following rectangle:

222555AAABBBDDDCCC

Find the area of the rectangle.

Video Solution

Step-by-Step Solution

Let's calculate the area of the rectangle by multiplying the length by the width:

2×5=10 2\times5=10

Answer

10

Exercise #10

Given the following rectangle:

111111777AAABBBDDDCCC

Find the area of the rectangle.

Video Solution

Step-by-Step Solution

Let's calculate the area of the rectangle by multiplying the length by the width:

11×7=77 11\times7=77

Answer

77

Question 1

Calculate the area of the following parallelogram:

666555

Question 2

Given that the diameter of the circle is 7 cm

What is the area?

777

Question 3

O is the center of the circle in the diagram below.

What is its area?

333OOO

Question 4

What is the area of the trapezoid in the figure?

777151515222AAABBBCCCDDDEEE

Question 5

ACBD is a deltoid.

AD = AB

CA = CB

Given in cm:

AB = 6

CD = 10

Calculate the area of the deltoid.

666101010AAACCCBBBDDD

Exercise #11

Calculate the area of the following parallelogram:

666555

Video Solution

Step-by-Step Solution

To solve the exercise, we need to remember the formula for the area of a parallelogram:

Side * Height perpendicular to the side

We can identify that in the diagram, although it's not presented in the way we're familiar with, we are given the two essential pieces of information -

Side = 6

Height = 5

Let's substitute into the formula and calculate:

6*5=30

And that's the solution!

Answer

30 cm²

Exercise #12

Given that the diameter of the circle is 7 cm

What is the area?

777

Video Solution

Step-by-Step Solution

First we need the formula for the area of a circle:

 πr2 \pi r^2

In the question, we are given the diameter of the circle, but we still need the radius.

It is known that the radius is actually half of the diameter, therefore:

r=7:2=3.5 r=7:2=3.5

We substitute the value into the formula.

π3.52=12.25π \pi3.5^2=12.25\pi

Answer

12.25π 12.25\pi cm².

Exercise #13

O is the center of the circle in the diagram below.

What is its area?

333OOO

Video Solution

Step-by-Step Solution

Remember that the formula for the area of a circle is

πR²

 

We insert the known data:

π3²

π9

 

Answer

9π 9\pi cm²

Exercise #14

What is the area of the trapezoid in the figure?

777151515222AAABBBCCCDDDEEE

Video Solution

Step-by-Step Solution

We use the following formula to calculate the area of a trapezoid: (base+base) multiplied by the height divided by 2:

(AB+DC)×BE2 \frac{(AB+DC)\times BE}{2}

(7+15)×22=22×22=442=22 \frac{(7+15)\times2}{2}=\frac{22\times2}{2}=\frac{44}{2}=22

Answer

22 22 cm².

Exercise #15

ACBD is a deltoid.

AD = AB

CA = CB

Given in cm:

AB = 6

CD = 10

Calculate the area of the deltoid.

666101010AAACCCBBBDDD

Video Solution

Step-by-Step Solution

To solve the exercise, we first need to remember how to calculate the area of a rhombus:

(diagonal * diagonal) divided by 2

Let's plug in the data we have from the question

10*6=60

60/2=30

And that's the solution!

Answer

30

Topics learned in later sections

  1. Area of a square