Circles and Pi:

Area of a Circle

If area is how many unit squares fit within a shape, how do we fit squares in a circle? Because squares don’t fit evenly into circles, we don’t usually get an even answer when we’re looking for the area of a circle. We also need to use \({\text{π}}\) (pi). The following video explains how to find the area of a circle from the radius.

Video Source (06:44 mins) | Transcript

Let \({\text{r}} = \text{radius}\)

\( \text{Area} = {\text{π}} {\text{ r}}^{2}\)

Standard Mathematical Formats with Pi

When pi is part of a solution there are two ways you can display the solution. The first way is to write the number part of the solution multiplied to pi such as 13\(\pi\) ft or 5.3\(\pi\) cm. We generally write the number then pi and then the units.

The second way to show your solution is to multiply the number portion of the solution to pi and then round to an appropriate place value. (Example: 13\(\pi\) ft = 40.84 ft rounded to the nearest hundredth)

In this course, we will always multiply pi into our solution and round to an appropriate place value. Just know, that the other way is commonly used and you may see it in textbooks or other classes as a standard way to write solutions when pi is involved.

Additional Resources

Khan Academy: Area of a Circle (4:01 mins; Transcript)
Khan Academy: Area of a Circle Intuition (9:25 mins; Transcript)

Practice Problems

A circle has a radius of 21.5. Find the area of the circle. Round to the nearest tenth.

A button is in the shape of a circle. If the radius of the button is 7 mm, find the surface area of the top of the button. Round to the nearest hundredth.

A helicopter landing pad has a diameter of 28 meters or a radius of 14 meters. Find the surface area of the landing pad. Round your answer to the nearest whole number.

A rock is thrown into a pond and creates circular ripples that radiate away from the center. At one point, the largest ripple has a radius of 7.25 feet. Find the surface area of the ripple. Round to the nearest tenth.

A frisbee is a circular disk tossed back and forth between players in a game. If the radius of the frisbee is 5 inches, find the surface area of the top of the frisbee. Round to the nearest tenth.

A cellphone tower has the ability to provide service to a circular region with a radius of 40.3 miles. Find the total surface area of the coverage zone. Round to the nearest hundredth.

Solutions

\(1452.2\) (when using the pi button on the calculator)
\(153.94 \text{ mm}^{2}\) (when using the pi button on the calculator)
\(616\:{\text{m}}^{2}\) (when using the pi button on the calculator) (Written Solution | Solution Video | Transcript)

We are trying to find the area of the helicopter pad, which is shaped like a circle. Since the pad is circular, we can use the formula for the area of a circle:

\({\text{A}} = {\text{π}} {\text{ r}}^{2}\)

The first thing we need to do is replace ‘r’ with 14m:

\({\text{A}} = {\text{π}} (14{\text{m}})^{2}\)

Next, we square 14 m (or multiply 14 x 14) to get \(196 {\text{m}}^{2}\) :

\({\text{A}} = {\text{π}} 196{\text{m}}^{2}\)

Since we can multiply in any order, we can switch the order of 196 and \({\text{π}}\) and write the area like this:

\({\text{A}} = 196 {\text{π}} {\text{ m}}^{2}\)

To write the area as a decimal, we multiply 196 by \(\pi\) to get:

\({\text{A}} = 615.7521601... {\text{m}}^{2}\)

So the area of the helicopter landing pad is about \(616 {\text{m}}^{2}\) when rounded to the nearest whole number.
\(165.1 \text{ ft}^{2}\) (when using the pi button on the calculator) (Written Solution)

We are trying to find the area inside the largest ripple. Since the ripples are circular, we can use the formula for the area of a circle:

\({\text{A}} = {\text{π}} {\text{ r}}^{2}\)

First, we replace ‘r’ with 7.25 ft

\({\text{A}} = {\text{π}} (7.25 {\text{ft}})^{2}\)

Next, we square 7.25 ft to get:

\({\text{A}} = {\text{π}} 52.5625 {\text{ft}}^{2}\)

We can switch the order of 52.5625 and \({\text{π}}\) since we can multiply in any order.

\({\text{A}} = 52.5625 {\text{π}} {\text{ ft}}^{2}\) (This is in a standard mathematical format.)

If needed, we can multiply \({\text{π}}\) by 52.5625 to get:

\({\text{A}} = 165.129963854... {\text{ft}}^{2}\)

So the area of the circular area inside the ripple is about \(165.1 {\text{ft}}^{2}\) when rounded to the nearest tenth.
\(78.5 \text{ in}^{2}\) (when using the pi button on the calculator)
\(5102.23 \text{ mi}^{2}\) (when using the pi button on the calculator) (Solution Video | Transcript)