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Circle math

Area and circumference of a circle

# 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…

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.

## 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.

## How to calculate the area or circumference of a circle

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

1. Find the radius of a circle or the diameter of a circle.
2. Use the relevant formula.

## 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.

1. 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}

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.

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

Use the relevant formula.

### Example 3: circumference of a circle given the radius

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

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

Use the relevant formula.

### Example 4: circumference of a circle given the diameter

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

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

Use the relevant formula.

### Example 5: area of a semicircle

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

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

Use the relevant formula.

### 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.

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

Use the relevant formula.

### 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

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.

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

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

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}

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

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}

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

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}

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

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}

5. Find the area of this quarter circle. Give your answer to 3 significant figures.

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}

6. The circumference of a circle is 50cm . Find the area of the circle. Give your answer to 2 decimal places.

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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