A sector of a circle of radius 24m has an area of 288m2. Find the central angle of the sector

The area \(A\) of a sector of a circle is given by the formula:

\[
A = \frac{\theta}{360^\circ} \times \pi r^2
\]

where:
- \(A\) is the area of the sector,
- \(\theta\) is the central angle of the sector in degrees,
- \(r\) is the radius of the circle.

### Step 1: Plug in known values
Given:
- \(A = 288 \, \text{m}^2\),
- \(r = 24 \, \text{m}\).

Now substitute these values into the formula:

\[
288 = \frac{\theta}{360^\circ} \times \pi (24)^2
\]

\[
288 = \frac{\theta}{360^\circ} \times \pi \times 576
\]

\[
288 = \frac{\theta \cdot 576\pi}{360^\circ}
\]

### Step 2: Solve for \(\theta\)
Rearrange the equation to isolate \(\theta\):

\[
288 \times 360^\circ = \theta \cdot 576\pi
\]

\[
103680^\circ = \theta \cdot 576\pi
\]

\[
\theta = \frac{103680^\circ}{576\pi}
\]

Using \(\pi \approx 3.1416\), we get:

\[
\theta = \frac{103680}{1809.56} \approx 57.3^\circ
\]

### Final Answer
The central angle of the sector is approximately:

\[
\boxed{57.3^\circ}
\]

Would you like a step-by-step breakdown of any part of this solution?

---

Here are 5 related questions for further practice:
1. What is the length of the arc corresponding to this sector?
2. How do you find the perimeter of a sector, given its radius and central angle?
3. If the area of a sector is doubled, how does the central angle change?
4. What is the area of a sector with a radius of 10 m and a central angle of 45°?
5. How does the formula for the area of a sector differ when the angle is in radians?

**Tip:** Always check if the angle is in degrees or radians before applying sector-related formulas.

A sector of a circle of radius 24m has an area of 288m². Find the central angle of the sector.

Solve the problem of finding the central angle of a sector with a radius of 24 meters and an area of 288 square meters. This solution explains the key geometric formula used to calculate the central angle and provides a step-by-step breakdown. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation