# Lesson 9

Applying Area of Circles

Let’s find the areas of shapes made up of circles.

### 9.1: Still Irrigating the Field

The area of this field is about 500,000 m^{2}. What is the field’s area to the nearest square meter? Assume that the side lengths of the square are exactly 800 m.

- 502,400 m
^{2} - 502,640 m
^{2} - 502,655 m
^{2} - 502,656 m
^{2} - 502,857 m
^{2}

### 9.2: Comparing Areas Made of Circles

- Each square has a side length of 12 units. Compare the areas of the shaded regions in the 3 figures. Which figure has the largest shaded region? Explain or show your reasoning.
- Each square in Figures D and E has a side length of 1 unit. Compare the area of the two figures. Which figure has more area? How much more? Explain or show your reasoning.

Which figure has a longer perimeter, Figure D or Figure E? How much longer?

### 9.3: The Running Track Revisited

The field inside a running track is made up of a rectangle 84.39 m long and 73 m wide, together with a half-circle at each end. The running lanes are 9.76 m wide all the way around.

What is the area of the running track that goes around the field? Explain or show your reasoning.

### Summary

The relationship between \(A\), the area of a circle, and \(r\), its radius, is \(A=\pi r^2\). We can use this to find the area of a circle if we know the radius. For example, if a circle has a radius of 10 cm, then the area is \(\pi \boldcdot 10^2\) or \(100\pi\) cm^{2}. We can also use the formula to find the radius of a circle if we know the area. For example, if a circle has an area of \(49 \pi\) m^{2} then its radius is 7 m and its diameter is 14 m.

Sometimes instead of leaving \(\pi\) in expressions for the area, a numerical approximation can be helpful. For the examples above, a circle of radius 10 cm has area about 314 cm^{2}. In a similar way, a circle with area 154 m^{2} has radius about 7 m.

We can also figure out the area of a fraction of a circle. For example, the figure shows a circle divided into 3 pieces of equal area. The shaded part has an area of \(\frac13 \pi r^2\).

### Glossary Entries

**area of a circle**If the radius of a circle is \(r\) units, then the area of the circle is \(\pi r^2\) square units.

For example, a circle has radius 3 inches. Its area is \(\pi 3^2\) square inches, or \(9\pi\) square inches, which is approximately 28.3 square inches.

**squared**We use the word

*squared*to mean “to the second power.” This is because a square with side length \(s\) has an area of \(s \boldcdot s\), or \(s^2\).