# The Area of a Circle

## Area and Circumference of a Circle – Definition

Circles are everywhere in our daily lives, from the wheels on a bus to the pizzas we enjoy. Understanding the area and circumference of a circle is not just a fundamental aspect of geometry, but also a practical skill used in various real-world scenarios.

The area of a circle measures the space inside it, while the circumference is the distance around the circle. These are calculated using the circle's radius or diameter and the constant Pi ($\pi$).

The key to working with circles is knowing their radius (the distance from the center to the edge) and diameter (the distance across the circle, passing through the center). The diameter is always twice the radius.

Concept Measurement Description Formula
Area of a Circle Measures the space inside the circle $A = \pi r^2$
Circumference of a Circle Measures the distance around the circle $C = 2\pi r$ or $C = \pi d$

## Steps to Find the Circumference of a Circle

Find the circumference of a circle with a radius of $4$ cm.

• Identify the radius: The radius of our circle is $4$ cm.

• Use the circumference formula: The formula for the circumference of a circle is $C = 2\pi r$.

• Substitute in the value for the radius: Substitute the radius into the formula: $C = 2\pi \times 4$.

• Calculate: Multiply $2$ by $\pi$ and then by $4$ to get the circumference.

• Result: The circumference of the circle is $8\pi$ cm, or approximately $25.13$ cm.

### Converting Between Radius and Diameter

• To find the diameter if you know the radius, multiply the radius by $2$.
• To find the radius if you know the diameter, divide the diameter by $2$.
If the radius of a circle is $5$ yards, what is the diameter?
A circle has a diameter of $12$ in. What is the radius of the circle?
If a bicycle wheel has a radius of $35$ cm, what is the diameter of the wheel?

## Area and Circumference of a Circle – Exercises

Practice finding the area and circumference of circles!

Calculate the area and then the circumference (in terms of $\pi$) of a circle with a radius of 6 cm.
Calculate the area and then the circumference of a circle with a diameter of 10 inches.
Calculate the area and then the circumference of a circle with a radius of 3.5 ft.
Calculate the area and then the circumference of a circle (in terms of $\pi$) with a diameter of 8 yards.
Calculate the area and then the circumference of a circle with a radius of 7 miles.

This is just the beginning of the use of pi, and there is so much more to learn all about pi!

## Area and Circumference of a Circle – Real-World Application

Knowing the area and circumference of a circle is practical for everyday tasks. Use the area for covering spaces like painting circles or gardening. Circumference helps when measuring around objects, like fencing a garden or sizing a table edge.

A circular track has a diameter of 400 meters. How long is one lap around the track?

Finding the parts of areas and circumferences of circles is a good next step to challenge yourself after feeling confident with the area and circumference of a circle.

## Area and Circumference of a Circle – Summary

Key Learnings from this Text:

• The area of a circle is found with $A = \pi r^2$.
• The circumference is calculated as $C = 2\pi r$ or $C = \pi d$.
• Understanding these concepts is crucial for real-world applications like planning and construction.
• Remember to double the radius to get the diameter, and halve the diameter to find the radius.

Not only is it important to know how to find the area and circumference of a circle, but so is understanding the relationship between circles and the number Pi.

## Area and Circumference of a Circle – Frequently Asked Questions

How do you find the area of a circle?
What is the formula for the circumference of a circle?
Can you calculate the circumference if you only know the area?
What is Pi ($\pi$) in circle calculations?
Why do we multiply the radius by $2$ to find the diameter?
Are the area and circumference of a circle always in the same units?
How does the diameter affect the area and circumference of a circle?
Can the area of a circle be more than its circumference?
What real-world applications use circle area and circumference calculations?
How do you calculate the radius from the circumference?

## The Area of a Circle exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the learning text The Area of a Circle .
• ### Demonstrate your understanding of the area of a circle.

Hints

The radius or $r$ of this circle is $7$ cm.

The diameter or $d$ of this circle is $14$ cm.

The radius is half of the diameter.

As long as the $r$ (radius) is given, it is possible to find the area of a circle. The radius can be substituted in the formula for $r$.

$r=7$ cm

$A=\pi r^2$

$A=\pi (7^2)$

Solution

The circle has a radius of 10 feet which is half of 20 feet. The green line is the diameter of the circle. To find the area of this circle, the formula used is $\bf{A= \pi r^2}$.

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The distance from the center of the circle to the outside is known as the radius and in this circle is 5 feet. Therefore the diameter is double, or 10 feet. Using the formula $A= \pi r^2$, the $5$ can replace the $r$ in the formula like this, $\bf{A= \pi (5^2)}$. This can then be calculated to find the approximate area of the circle.

• ### Determine the steps used to find the area of a circle.

Hints

The radius of a circle, $r$ is the distance from the center of a circle to the outside.

When a number is squared ($g^2$), it means that the number is multiplied by itself ($g \times g$).

For example, $8^2=8\times 8$.

After squaring the radius ($r=4$) in the formula,

$A=4^2 \pi$

$A=16 \pi$

multiply the value by $\pi$ using the button for $\pi$ on a calculator.

$16 \times \pi = 50.2654825...$

When giving a value of a 2D shape's area, square units are always used.

Some examples include: $\text{cm}^2$, $\text{in}^2$ or $\text{ft}^2$.

Solution

Step 1

Identify the formula needed to find the area of a circle.

$A=\pi r^2$

${ }$

Step 2

Determine the measurement of the circle's radius and substitute that value into the formula.

$r=5$

$A=\pi (5^2)$

${ }$

Step 3

$5^2=25$

$A=25 \pi$

${ }$

Step 4

Calculate the radius squared multiplied by $\pi$.

$25 \times \pi = 78.539...$

${ }$

Step 5

Round your answer as directed, and include the appropriate units for the area of the circle.

$A \approx 78.5 \: \text{ft}^2$

• ### Use a formula to find the area of a circle.

Hints

If you know the radius, this value can be substituted in the formula for $r$.

Here is a an example of how to find the area if we had a radius of 11 inches.

$r = 11\:\text{in}$

$A = \pi r^2$

$A= \pi (11^2)$

$A \approx 380.132711...$

$A \approx 380 \:\text{in}^2$

To round to the nearest square inch, it is the same as rounding to the nearest whole number, or ones digit.

Remember to be sure to include your squared units at the end of the measurement.

Instead of $\text{in}$, use $\text{in}^2$.

Solution

$r = 12\:\text{in}$

$A = \pi r^2$

$A= \pi (12^2)$

$A \approx 452.389342...$

$A \approx 452 \:\text{in}^2$

• ### Apply the formula for the area of a circle.

Hints

The $A$ is the area, and the $r$ is the radius of the circle.

This example shows the steps taken to find the area of a circle.

Radius: The radius of a circle is like a straight line going from the very center of the circle to its edge. It's like the distance from the middle of a pizza to its crust.

Diameter: The diameter of a circle is a straight line that goes from one side of the circle to the other side, passing through the center. It's like measuring a pizza straight across from one edge to the opposite edge. The diameter is always two times longer than the radius.

Always be sure to check if you are given the radius or the diameter.

If given the diameter, you need to halve the value to find the radius.

For example, if we were given a diameter of 22 inches, half of 22 is 11, so the radius is 11 inches.

Solution

$r=8\:\text{cm}$ ; $A \approx 201.1\:\text{cm}^2$

$r=4\:\text{cm}$ ; $A \approx 50.3\:\text{cm}^2$

$d=8.5\:\text{cm}$ ; $A \approx 56.7\:\text{cm}^2$

$d=9\:\text{cm}$ ; $A \approx 63.6\:\text{cm}^2$

• ### Understand how to apply the formula for the area of a circle.

Hints

This is the formula used to find the area of a circle.

$A$ = Area

$r$ = radius

$r^2$ means $r \times r$

For example if $r=3$, $3^2 = 9$.

The radius in this circle is $15\:m$.

The formula used to find the area would be $A=\pi \times 15^2$.

Solution

$r=9$ meters

$A= \pi (r^2)$

Substitute the value for $r$ into the formula.

$A= \pi (9^2)$

• ### Demonstrate your knowledge for calculating the area of a circle.

Hints

The formula for the area of a circle is $A=\pi r^2$.

$r$ = radius

$A$ = Area

For this example, we only need the area of the donut shape. Subtraction would be helpful to remove the area not needed from the hole of the donut.

Steps to find the area of the donut:

• Find the area of the large circle.
• Find the area of the smaller inner circle.
• Subtract them to find just the area of what is asked.
Solution

Area of the entire circle:

$A=\pi (7^2)$

$A \approx 154\:\text{cm}^2$

Area of the inner circle:

$A=\pi (2^2)$

$A \approx 13\:\text{cm}^2$

Subtract the values to just find what is being asked.

$154 - 13 = 141$

$A \approx 141 \:\text{cm}^2$

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