Area And Circumference of a Circle - Math Steps with Examples

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Area and circumference of a circle

Here you will learn about calculating the area and circumference of a circle, including using the area of a circle and circumference of a circle formulas.

Students will first learn about the area and circumference of a circle in $7$ th grade geometry and expand on their knowledge in high school.

What is the area and circumference of a circle?

The area of a circle is the amount of space within a circle and the circumference of a circle is the distance around the edge of the circle.

To calculate the area and circumference of a circle, you use the following circle formulas:

$Area = \pi r^{2}$

$Circumference = \pi d = 2\pi r$

Where $r$ is the value of the radius of the circle (the distance from the center of the circle to the edge) and $d$ is the diameter of the circle (the distance across the circle passing through the center of the circle).

The symbol $\pi$ is the Greek letter “pi” where the value of pi is $3.14159265358979\dots$

As $\pi$ is an irrational number, the approximate value of $\pi$ is $3.142 (3dp).$

The radius of a circle is a line segment that runs from the center of a circle to any point on the perimeter of a circle, or circumference.

The diameter is a straight line that passes through the center of a circle and two points on the perimeter of a circle, or circumference.

The circumference of the circle is the perimeter of a circle and is proportional to its radius; as the radius $r$ increases, the circumference increases.

The circumference is a length and is measured using units, such as $mm, \; cm, \; m.$

Area is measured in square units, such as $mm^2, cm^2$ or $m^2.$

What is the area and circumference of a circle?

Common Core State Standards

How does this relate to $7$ th grade math?

Grade 7 – Geometry (7.G.B.4)
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

[FREE] Area And Circumference Of A Circle Worksheet (Grade 7)

Use this worksheet to check your 7th grade students’ understanding of area and circumference of a circle. 15 questions with answers to identify areas of strength and support!

DOWNLOAD FREE

[FREE] Area And Circumference Of A Circle Worksheet (Grade 7)

Use this worksheet to check your 7th grade students’ understanding of area and circumference of a circle. 15 questions with answers to identify areas of strength and support!

DOWNLOAD FREE

How to calculate the area or circumference of a circle

In order to calculate the area or circumference of a circle:

Find the radius of a circle or the diameter of a circle.
Use the relevant formula.
Give your answer clearly with the correct units.

Area and circumference of a circle examples

Example 1: area of a circle given the radius

Find the area of this circle. Give your answer to $1$ decimal place.

Find the radius of a circle or the diameter of a circle.

To find the area, you need to know the radius. The radius of this circle is $9~{cm}.$

2Use the relevant formula.

The circle formula for area is $A= \pi r^{2}.$

\begin{aligned} A&= \pi r^{2} \\\\ A&= \pi \times 9^{2} \\\\ &=254.4690049 \\\\ \end{aligned}

3Give your answer clearly with the correct units.

You need to give the answer to $1$ decimal place. Since the radius is measured in $cm,$ the area will be measured in $cm^2.$

Area =254.5 \mathrm{~cm}^2

Example 2: area of a circle given the diameter

Find the area of this circle. Give your answer in terms of $\pi.$

To find the area, you need to know the radius. The diameter of this circle is $6~{mm}$ , and therefore the radius is $3~{mm}.$

The formula for area is $A= \pi r^{2}.$

\begin{aligned} A&= \pi r^{2} \\\\ A&= \pi \times 3^{2} \\\\ &=9 \pi \end{aligned}

Here, you are asked to give the answer in terms of pi $(\pi).$ The diameter is measured in $mm$ so the area will be measured in $mm^2.$

$Area = 9 \pi~{mm}^{2}$

Example 3: circumference of a circle given the radius

Find the circumference of this circle. Give your answer to $3$ significant figures.

To find the circumference, you can use the radius or the diameter. The radius of this circle is $21~{m}.$

To find the circumference using the radius, you can use the formula $C=2 \pi r.$

\begin{aligned} & C=2 \pi r \\\\ & C=2 \times \pi \times 21 \\\\ & C=131.9468915 \end{aligned}

Here, you are asked to give the answer to $3$ significant figures. The radius is measured in $m$ so the circumference will also be measured in $m.$

$Circumference =132~{m} \; (3sf)$

Example 4: circumference of a circle given the diameter

Find the circumference of this circle. Give your answer in terms of $\pi.$

To find the circumference, you can use the radius or the diameter. The diameter of this circle is $17~{cm}.$

To find the circumference using the diameter, you can use the formula $C= \pi d.$

\begin{aligned} & C=\pi d \\\\ & C=\pi \times 17 \\\\ & C=17 \pi \end{aligned}

Here, you are asked to give the answer in terms of $\pi.$ The diameter is measured in $cm$ so the circumference will also be measured in $cm.$

$Circumference =17 \pi ~ \mathrm{cm}$

Example 5: area of a semicircle

Find the area of this semicircle. Give your answer to $2$ decimal places.

From the diagram of the semicircle, you can see that the diameter of the circle is $10~{cm}.$ For area, you need the radius, which is $10 \div 2 =5~{cm}.$

To find the area of a semicircle, you find the area of the whole circle and half it.

\begin{aligned} A&=\pi{r}^{2}\div{2} \\\\ &=\pi\times{5}^{2}\div{2} \\\\ &=25\pi\div{2} \\\\ &=12.5\pi \\\\ &=39.26990817 \end{aligned}

You have been asked to give the answer to $2$ decimal places. The diameter is measured in $cm$ therefore the area will be measured in $cm^2.$

$Area =39.27cm^{2} \; (2dp)$

Example 6: circumference of a circle given the area

The area of a circle is $30~{cm}^2.$ Find the circumference of the circle. Give your answer to $3$ significant figures.

In this case, you are given the area of the circle and will work backwards to find the radius. Substituting the values you know into the area formula, you get:

\begin{aligned} \text{Area }&= \pi r^{2} \\\\ 30&= \pi r^{2} \end{aligned}

Then solve to find $r.$

$30=\pi{r}^{2}$

Dividing both sides by $\pi,$ you get

$\cfrac{30}{\pi}=r^{2}$

As you have $r^2,$ you need to square root both sides to get $r$ on its own.

$\sqrt{\cfrac{30}{\pi}}=r$

Calculating this, you have $r=3.090193616.$

It is important not to round this value yet as you will lose accuracy in your final answer.

Now that you have the radius, use the relevant formula to find the circumference.

\begin{aligned} C&=2 \pi r \\\\ &=2 \times \pi \times 3.090193616 \\\\ &=19.41625913 \end{aligned}

You have been asked to give the answer to $3$ significant figures. The area is measured in $cm^2$ , so the circumference will be measured in $cm.$

$Circumference =19.4 \mathrm{~cm} \; (3sf)$

Teaching tips for area and circumference of a circle

For students that are struggling, connect the concepts of area and circumference to real-world scenarios, such as calculating the area of a pizza or the circumference of a circular track.

Use visual aids, diagrams, and interactive demonstrations to illustrate the formulas for area $(A= \pi r^2)$ and circumference $(C=2 \pi r).$ Visualizing the relationships between radius, diameter, and these formulas helps students build their understanding of the concepts.

While worksheets have their place when practicing finding the area and circumference, emphasize breaking down problem-solving into step-by-step procedures. Make sure to guide students through the process of applying the formulas, and correctly identifying and using the radius or diameter in each context.

Easy mistakes to make

Confusing diameter and radius
Students might accidentally use the diameter in the formulas for area $(A= \pi r^2)$ or circumference $(C=2 \pi r)$ instead of the correct radius. This error may lead to incorrect answers.

Not squaring the radius when using formulas
A common oversight is forgetting to square the radius when calculating the area of a circle $(A = \pi r^2).$ Students may mistakenly use the formula $A = \pi r$ instead.

Incorrectly rounding answers
Rounding errors can occur when students round off intermediate values too early in the calculations. This can lead to inaccuracies in the final area or circumference result, especially if rounded values are used in subsequent steps.

Using the wrong formula
Confusing the formulas for area and circumference is a frequent mistake. Students might inadvertently use the formula for circumference $(C=2 \pi r)$ when calculating the area, or vice versa, resulting in inaccurate answers.

Practice area and circumference of a circle questions

69.1 \mathrm{~in}^2

1,520.5 \mathrm{~in}^2

34.6 \mathrm{~in}^2

380.1 \mathrm{~in}^2

The radius of the circle is $11~{cm}.$

\begin{aligned} \text { Area }&=\pi r^2\\\\ & =\pi \times 11^2 \\\\ & =380.1327111 \\\\ & =380.1 \mathrm{~cm}^2 \; (1 \mathrm{dp}) \end{aligned}

49 \pi \mathrm~{mm}^{2}

14 \pi \mathrm~{mm}^{2}

28 \pi \mathrm~{mm}^{2}

196 \pi \mathrm~{mm}^{2}

The diameter is $14~{mm}$ so the radius is $7~{mm}.$

\begin{aligned} \text{Area }&= \pi r^{2} \\\\ &= \pi \times 7^{2} \\\\ &=49 \pi ~ \mathrm{mm}^{2} \end{aligned}

2.51 \mathrm{~m}

2.01 \mathrm{~m}

5.03 \mathrm{~m}

8.04 \mathrm{~m}

The radius of the circle is $0.8~{m}.$

\begin{aligned} \text { Circumference }&=2 \pi r \\\\ & =2 \times \pi \times 0.8 \\\\ & =5.026548246 \\\\ & =5.03 \, m \; (3 s f) \end{aligned}

26 \pi ~\mathrm{cm}

13 \pi ~\mathrm{cm}

169 \pi ~\mathrm{cm}

676 \pi ~\mathrm{cm}

The diameter of the circle is $26~{cm}.$

\begin{aligned} \text{Circumference }&=\pi d\\\\ &=\pi \times 26\\\\ &=26 \pi~\mathrm{cm} \end{aligned}

1,020 \mathrm{~cm}^2

254 \mathrm{~cm}^2

28.3 \mathrm{~cm}^2

63.3 \mathrm{~cm}^2

You are finding the area of $\cfrac{1}{4}$ of a circle. The diagram shows that the radius of the circle is $18~{cm}.$

Find the area of the whole circle and then divide by $4.$

\begin{aligned} \text{Area of whole circle }&=\pi r^{2}\\\\ &=\pi \times 18^{2}\\\\ &=324 \, \pi \end{aligned}

\begin{aligned} \text{Area of quarter circle } =324\pi \div 4&= 81\pi\\\\ &=254.4690049\\\\ &=254 \, \mathrm{cm}^{2}~\text{(3sf)} \end{aligned}

1,963.50 \mathrm{~cm}^2

15.92 \mathrm{~cm}^2

795.77 \mathrm{~cm}^2

198.94 \mathrm{~cm}^2

To find the area of the circle, you will first need to find the diameter of the circle by working backwards.

\begin{aligned} \text{Circumference }&=\pi d\\\\ 50&=\pi d\\\\ \frac{50}{\pi}&=d\\\\ d&=15.91549431\\ \end{aligned}

\begin{aligned} r&=15.91549431 \div 2\\\\ &=7.957747155 \end{aligned}

Next, use the radius to find the area of the circle.

\begin{aligned} \text{Area }&=\pi r^{2}\\\\ &=\pi \times 7.957747155^{2}\\\\ &=198.9436789\\\\ &=198.94 \; \mathrm{cm}^{2} \; (2dp) \end{aligned}

Area and circumference of a circle FAQs

What is the formula for the area of any circle, and how is it calculated?

The area of a circle formula is $A=\pi r^2,$ where $r$ is the radius. To calculate the area, square the radius and multiply it by the mathematical constant $\pi$ (pi).

How can I find the circumference of any circle, and what is the formula?

There are two formulas that can be used to find the circumference of circles. One circumference formula is $C=2 \pi r,$ where $r$ is the radius, and the second circumference formula is $C=\pi d,$ where $d$ is the diameter.

How do you find the radius of a circle?

The radius is a line segment from the center of a circle to any point on its circumference. If the diameter of a circle is given, you can simply divide the diameter by $2$ to find the radius.

What is the difference between finding the area and finding the surface area of a figure?

Area refers to the total space of a $2D$ shape, such as a circle, and surface area refers to the total space of a $3D$ shape. For example, you would find the area of a square, area of a rectangle, or area of a triangle, but would find the surface area of a cube or rectangular prism, and triangular pyramid.

The next lessons are

Sectors arcs and segments
Angles of a circle
Circle theorems
Prism shape

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