Square Inside a Circle Area
Support Page

Welcome to our Square Inside a Circle Area support page.

Looking for some help and support to find the area of a square inside a circle?

Then look no further - we have explanations, worked examples and practice sheets to help you master this skill!

Square Inside a Circle Area Intro

This page is all about how to find the area of a square whose sides are all touching the edge of a circle, as shown below.

square inside a circle ex 1

We can use the information on this page to find the area of the square inside the circle (shaded green - as in Examples A and C).

We can also use the information to find the area of the circle that is outside of the square by subtracting the area of the square from the area of the circle (as in example B).

On this webpage you will find a range of support, examples and practice worksheets to help your child learn to work out the area of a square within a circle, or the left-over area of a circle outside a square (Example B).

A square inside a circle (as shown) is sometimes called a square inscribed by a circle.

Our Square Inside a Circle Area sheets are graded from easiest to hardest, and each sheet comes complete with answers.

Square Inside a Circle Area Formula

Formula for finding the Area of a Square inside a Circle

For those of you who just like to know what the formula is:

Area of Square Inside a Circle = 2r2

square inside a circle area formula

Where r is the radius of the circle, and also the distance from the center of the square to one of its corners.


Finding the Area of the Circle that is not inside the square (the part of the circle shaded green below).

square inside a circle example 8

The formula for the area of a circle is A = πr2

To find the area of the part of the circle not inside the square, we just subtract the area of the square from the area of the circle.

= πr2 - 2r2 = (π -2)r2

So the Area of the Circle left over is (π -2)r2

Square Inside a Circle Area - How it Works!

For those of you who like to know how things work, and why the formula is 2r2, we will show you how it can be found below!

Let's start by looking at the four corners of the square and draw in one of the diagonals.

square inside a circle ex 2

You will notice that the diagonal of the square is also the diameter of the circle as the diagonal goes from one side of the circle to the other through the center.

This means that the length of the diagonal is 2r, where r is the radius of the circle.

You can also see that the diagonal and two sides of the square into two right triangles.

square inside a circle ex 3

Let's look closely at the right triangle that has been made by the diagonal of the square:

square inside a circle ex 4

As the triangle is a right triangle, we can use Pythagoras' theorem to find the missing side s.

Pythagoras' theorem states that the sum of the squares of the two smaller sides of the triangle is equal to the square of the hypotenuse (which in this case is equal to 2r).

So s2 + s2= (2r)2

This gives us 2s2 = 4r2

So s2 = 2r2

So s = √(2r2)

We can simplify this to s = √2 r

As s is the vaue for the side of the square, this gives us:

square inside a circle ex 5

The area of the square can now simply be found by multiplying two adjacent sides together to give:

√2 r x √2 r = √2 x √2 x r x r = 2r2

So the Area of the Square Inside the Circle is 2r2


We can also say that the perimeter of the square is 4 x √2 r = 4√2 r

Square Inside a Circle Area Examples

Area of a Square Inside a Circle Example 1

Find the area of the square below.

square inside a circle example 1

To find the area of the square we need to work out 2r2, where r=4cm.

42 = 4x4 = 16

So 2r2 = 2 x 16 = 32.

So the area of the square is 32 cm2.


Area of a Square Inside a Circle Example 2

Find the area of the square below.

square inside a circle example 2

In this case, the diameter of the circle is known, which is equal to twice the value of the radius.

So r = 12 ÷ 2 = 6 in

To find the area of the square we need to work out 2r2, where r=6 in.

62 = 6x6 = 36

So 2r2 = 2 x 36 = 72.

So the area of the square is 72 in2.


Area of a Square Inside a Circle Example 3

Find the area of the shaded parts of the circle that are outside square below. Give your answer to 1dp.

square inside a circle example 3

In this case, the we need to find the area of the parts of the circle that are outside of the square.

To do this we need to find the area of the square and subtract it from the area of the circle.

To find the area of the square we need to work out 2r2, where r = 7 cm.

72 = 7x7 = 49

So 2r2 = 2 x 49 = 98.

So the area of the square is 98 cm2.

Next we will find the area of the whole circle.

The area of a circle is π r2 = π 72 = 49 π = 153.9 cm2 to 1dp

So we need to find the area of the whole circle - the area of the square

= 159.9 cm2 - 98 cm2 = 55.9 cm2

So the area of the parts of the circle outside the square = 55.9 cm2


Area of a Square Inside a Circle Example 4

Find the diameter of the circle below, when the area of the square is given.

square inside a circle example 4

In this case, we know the area of the square, but we need to find the diameter of the circle.

We know that the area of the square is 2r2, where r is the radius of the circle.

This means that 2r2 = 162 cm2

So r2 = 162 ÷ 2 = 81 cm2.

So r = √81 = 9 cm

Now we have found the radius of the circle, we just need to double it to give us the diameter.

So the diameter of the circle is 9 x 2 = 18 cm.


Square Inside a Circle Area Worksheets

Area of a Square Inscribed by a Circle Worksheets

More Recommended Math Worksheets

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More Area Support and Resources

We have a range of other area worksheets and support pages for a range of different 2d shapes.

 

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