Question

In circle S with m, angle, R, S, T, equals, 120, degreesm∠RST=120
∘
and R, S, equals, 13RS=13, find the area of sector RST. Round to the nearest hundredth.

Answer 1

To find the area of sector RST in circle S, we can use the following formula for the area of a sector:

\[
\text{Area of sector} = \frac{m\angle}{360^\circ} \times \pi r^2
\]

Where:
- \( m\angle \) is the measure of the angle of the sector (in degrees),
- \( r \) is the radius of the circle.

Given:
- \( m\angle RST = 120^\circ \)
- \( R S = 13 \) (which represents the radius of the circle)

Now, substituting the values into the formula:

\[
\text{Area of sector} = \frac{120^\circ}{360^\circ} \times \pi (13)^2
\]

Calculating \( \frac{120^\circ}{360^\circ} \):

\[
\frac{120}{360} = \frac{1}{3}
\]

Next, calculate \( (13)^2 \):

\[
13^2 = 169
\]

Now substituting back into the area formula:

\[
\text{Area of sector} = \frac{1}{3} \times \pi \times 169
\]

This simplifies to:

\[
\text{Area of sector} = \frac{169\pi}{3}
\]

Now, let's calculate \( \frac{169\pi}{3} \):

Using \( \pi \approx 3.14159 \):

\[
\frac{169\pi}{3} \approx \frac{169 \times 3.14159}{3} \approx \frac{530.14101}{3} \approx 176.71367
\]

Rounding to the nearest hundredth:

\[
\text{Area of sector} \approx 176.71
\]

Thus, the area of sector RST is approximately **176.71 square units**.