Math resources Geometry

Circle math

Pi r squared

# 𝝅r² (pi r squared)

Here you will learn about using the formula \boldsymbol{\pi} r^2 \text { (pi r squared) } to calculate the area of a circle given the radius, diameter, or circumference.

Students will first learn about \boldsymbol{\pi} r^2 \text { (pi r squared) } as part of geometry in 7 th grade.

## What is pi r squared?

Pi r squared (pi times the radius squared) is the formula for the area of a circle.

This is because there is a specific relationship between the radius of a circle and its area.

Area of a circle formula:

For example,

What is the area of a circle with radius 5 \, cm? Give your answer to the nearest tenth.

Squaring the radius of 5 gives 25. Then multiply pi by this value to find the area of the circle.

\begin{aligned}\text { Area } & =\pi r^2 \\\\ & =\pi \times 5^2 \\\\ & =25 \pi \mathrm{~cm}^2 \\\\ & =78.5 \mathrm{~cm}^2 \text { (to the nearest tenth) }\end{aligned}

### What is pi?

\pi (pronounced pi) is a Greek letter that represents the ratio of the circumference of a circle to its diameter.

For all circles, if you divide the length of the circumference by the length of the diameter you get the value \boldsymbol{\pi}.

Note: \boldsymbol{\pi} is an irrational number which means it cannot be written as a fraction. It is a non-recurring decimal and has an approximate value of 3.1415…

Sometimes the question may ask you to give the answer ‘in terms of \pi ’.

For example,

3 \times \pi = 3\boldsymbol{\pi} (this is an answer in terms of pi)

5 \times \pi = 5\boldsymbol{\pi} (this is an answer in terms of pi)

17 \times \boldsymbol{\pi} = 53.407… (this is an answer not in terms of pi)

## 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 use pi r squared

In order to calculate the area of a circle:

1. Find the radius of the circle.
2. Use the formula \boldsymbol{\pi} r^2 to calculate the area of the circle.

## Pi r squared examples

### Example 1: calculating the area of the circle given the radius

A circle has a radius of 7 \, cm.

Calculate its area.

1. Find the radius of the circle.

The radius is given in the question.

2Use the formula \boldsymbol{\pi} r^2 to calculate the area of the circle.

\begin{aligned}& \pi r^2 \\\\ &\pi \times r\times r \\\\ &\pi \times7\times 7 \\\\ &49\pi \\\\ &= 153.93904\dots \end{aligned}

153.94

153.94 \, cm^2

### Example 2: calculating the area of the circle given the diameter

The diameter of a circle is 20 \, mm.

Calculate its area.

Find the radius of the circle.

Use the formula \boldsymbol{\pi} r^2 to calculate the area of the circle.

### Example 3: calculating the area of the circle, given the radius, answer in terms of π.

A circle has a radius of 18 \, m.

Calculate its area.

Find the radius of the circle.

Use the formula \boldsymbol{\pi} r^2 to calculate the area of the circle.

### Example 4: calculating the area of the circle given the diameter

A circle has a diameter of 210 \, km.

Calculate its area.

Find the radius of the circle.

Use the formula \boldsymbol{\pi} r^2 to calculate the area of the circle.

### Example 5: calculating the area of the circle given the circumference of a circle

A circle has a circumference of 12 \, cm.

Calculate its area.

Find the radius of the circle.

Use the formula \boldsymbol{\pi} r^2 to calculate the area of the circle.

### Example 6: calculating the area of a semicircle given the diameter

A semicircle has a diameter of 200 \, m.

Calculate its area.

Find the radius of the circle.

Use the formula \boldsymbol{\pi} r^2 to calculate the area of the circle.

### Teaching tips for 𝝅r² (pi r squared)

• Encourage students to draw circles, label the radius, and visualize how squaring the radius of the circle (r^2) relates to the circle’s area.

• Relate the concept to real-world examples where circles are prevalent, such as pizza, wheels, or clocks. Discuss how understanding the area of a circle is practical in various situations.

• Compare the formula for the area of a circle with other area formulas, like the area of a square or parallelogram. Highlight the unique characteristics of circles.

### Easy mistakes to make

You must have the radius to find the area of a circle using the area formula. A question may not give you the radius directly and so you need to use the information given to find the radius first.

• Not including the correct units
When working with area, you must always give the correct units squared
For example,
\mathrm{cm}^2, \mathrm{~m}^2, \mathrm{~km}^2 etc.

• Not rounding correctly
These questions often involve rounding. Only round at the end of the question and to the specified number of decimal places.

• Not giving answer in terms of \boldsymbol{\pi}
Sometimes the question may ask you to give the answer ‘in terms of \boldsymbol{\pi}’. This means you do not give the numerical answer that is produced when you multiply it by \boldsymbol{\pi}.
For example,
6 \times \pi=6 \boldsymbol{\pi} (this is an answer in terms of pi)
6 \times \pi=18.8495592 \ldots (this answer is not in terms of pi)

### Practice pi r squared questions

1. A circle has a diameter of 12 \, cm.  What is the radius of the circle?

6 \, cm

3 \, cm

12 \, cm

6\pi \, cm

The diameter of the circle is twice the size of the radius.

Therefore, to find the radius you can divide the diameter by 2.

12 \mathrm{~cm} \div 2=6 \mathrm{~cm}

2. What is the formula for the area of a circle?

\pi d

2\pi r

\pi r^2

\pi r
\pi \times r\times r

Simplify this to,

\text { Area }=\pi r^2

3. A circle has a radius of 2 \, cm. What is its area to the nearest tenth?

4\pi \, cm^2

4\pi \, cm

12.6 \, cm

12.6 \, cm^2

Area of a circle =\pi r^2

\pi \times 2 \times 2 is equal to 12.566…

This answer is correctly rounded to the nearest tenth and has the correct units.

4. A circle has a radius of 9 \, cm. What is its area in terms of \boldsymbol{\pi}?

9\pi \, cm^2

324\pi \, cm^2

18\pi \, cm^2

81\pi \, cm^2

Area of a circle =\boldsymbol{\pi} r^2

\begin{aligned}& \boldsymbol{\pi} \times 9 \times 9 \\\\ & 81 \boldsymbol{\pi} \, cm \end{aligned}

This answer is correctly given in terms of \boldsymbol{\pi}.

5. A circle has a diameter of 8 \, cm. What is its area in terms of \boldsymbol{\pi}?

16\pi \, cm^2

64\pi \, cm^2

8\pi \, cm^2

2\pi \, cm^2

8 \, cm = diameter

Area of a circle =\boldsymbol{\pi} r^2

\begin{aligned}& \pi \times 4 \times 4 \\\\ & 16 \pi \, cm \end{aligned}

This answer is correctly given in terms of \boldsymbol{\pi}.

6. A circle has a radius of 10 \, cm. What is its area to the nearest whole number?

314 \, cm^2

314.1 \, cm^2

314.159 \, cm^2

100\pi \, cm^2

Area of a circle =\boldsymbol{\pi} r^2

\pi \times 10 \times 10 is equal to 314.1592654…

This answer is correctly rounded to the nearest tenth and has the correct units.

## Pi r squared FAQs

What is a circle?

A circle is a shape formed by a curved line; each point on the line is equidistant from the center of the circle.

How do you find the area of circles?

To find the area of circles, use the formula area A=\boldsymbol{\pi} r^2 (pi r squared). Area is measured in square units (square meters, square inches, etc.)

Why do we use pi to find the area of circles?

We use pi (\pi) to find the area of circles because it represents the ratio of a circle’s circumference to its diameter. The formula for the area of a circle uses \pi to mathematically relate the size of a circle (measured by its radius r ) to the space it covers.

How can you use the formula A=\boldsymbol{\pi} r^2 (pi r squared) to find the radius of a circle?

You can use the formula A=\boldsymbol{\pi} r^2 (pi r squared) to find the radius of a circle if you know the area of the circle. Divide area A by \pi, and then find the square root of the quotient.

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