Question

Kelsey is given a circle with a radius of 5 cm. She calculates the area of a sector that has an angle of 216° and provides the solution of 3π cm2 . What was her mistake?

Answer 1

To find the area of a sector of a circle, we can use the following formula:

\[
\text{Area of sector} = \frac{\theta}{360°} \times \pi r^2
\]

where \(\theta\) is the angle of the sector in degrees, and \(r\) is the radius of the circle.

In this case, Kelsey has a radius \(r = 5 \, \text{cm}\) and an angle of \(\theta = 216°\). Let's plug these values into the formula:

1. First, calculate the area of the entire circle:

\[
\text{Area of circle} = \pi r^2 = \pi (5 \, \text{cm})^2 = 25\pi \, \text{cm}^2
\]

2. Now we calculate the area of the sector:

\[
\text{Area of sector} = \frac{216°}{360°} \times 25\pi \, \text{cm}^2
\]

3. Simplify the fraction:

\[
\frac{216}{360} = \frac{3}{5}
\]

4. Now calculate the area:

\[
\text{Area of sector} = \frac{3}{5} \times 25\pi \, \text{cm}^2 = 15\pi \, \text{cm}^2
\]

Thus, the correct area of the sector is \(15\pi \, \text{cm}^2\).

Kelsey's mistake was calculating the area of the sector as \(3\pi \, \text{cm}^2\), while it should actually be \(15\pi \, \text{cm}^2\). It seems she may have miscalculated the fraction or the product.