Sector Area Calculator

Published on: February 23, 2025
Final Answer: Free • Full Steps: Plus

This Sector Area Calculator finds the area of the slice of a circle cut out by an angle, and shows the working line by line. The angle has to be in radians before the formula behaves, so it is converted first, and the radius is then squared. Halving the product of those two pieces gives the area, which is the fraction of the whole circle that the angle opens up. Following the same pattern every time keeps the conversion and the squaring from being skipped.

Step-by-step method

  1. Set up the problem.
  2. Convert the angle to radians.
  3. Write the sector area formula.
  4. Substitute the radius and the angle in radians.
  5. Square the radius.
  6. Multiply to get the area.

Formulas:

\(A = \frac{1}{2} \times r^{2} \times \theta_{rad}\)
\(\theta_{rad} = \theta \times \frac{\pi}{180}\)
See Example 1 Hide Example 1

Problem

\(r = 7,\; \theta = 45^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Convert the angle to radians.
  3. Write the sector area formula.
  4. Substitute the radius and the angle in radians.
  5. Square the radius.
  6. Multiply to get the area.

Step 1

Step 1 - Set up the problem.

In this problem: The sector comes from a circle of radius \(r = 7\) and opens through \(\theta = 45^{\circ}\).

\(r = 7,\; \theta = 45^{\circ}\)

Step 2

Step 2 - Convert the angle to radians.

In this problem: Multiplying \(45^{\circ}\) by \(\frac{\pi}{180}\) gives \(0.79\) radians.

\(\theta_{rad} = \frac{45\pi}{180} \approx 0.79\)

Step 3

Step 3 - Write the sector area formula.

In this problem: A sector is the fraction of the circle that the angle cuts out, which gives \(A = \frac{1}{2} \times r^{2} \times \theta_{rad}\).

\(A = \frac{1}{2} \times r^{2} \times \theta_{rad}\)

Step 4

Step 4 - Substitute the radius and the angle in radians.

In this problem: Replacing \(r\) with \(7\) and \(\theta_{rad}\) with \(0.79\) gives \(A = \frac{1}{2} \times 7^{2} \times 0.79\).

\(A = \frac{1}{2} \times 7^{2} \times 0.79\)

Step 5

Step 5 - Square the radius.

In this problem: Squaring \(7\) gives \(49\).

\(A = \frac{1}{2} \times 49 \times 0.79\)

Step 6

Step 6 - Multiply to get the area.

In this problem: Multiplying \(49\) by \(0.79\) and halving gives \(19.24\).

\(A = \frac{1}{2} \times 49 \times 0.79 \approx 19.24\)

Final Answer

\(A \approx 19.24\)
See Example 2 Hide Example 2

Problem

\(r = 3,\; \theta = 150^{\circ}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Convert the angle to radians.
  3. Write the sector area formula.
  4. Substitute the radius and the angle in radians.
  5. Square the radius.
  6. Multiply to get the area.

Step 1

Step 1 - Set up the problem.

In this problem: The sector comes from a circle of radius \(r = 3\) and opens through \(\theta = 150^{\circ}\).

\(r = 3,\; \theta = 150^{\circ}\)

Step 2

Step 2 - Convert the angle to radians.

In this problem: Multiplying \(150^{\circ}\) by \(\frac{\pi}{180}\) gives \(2.62\) radians.

\(\theta_{rad} = \frac{150\pi}{180} \approx 2.62\)

Step 3

Step 3 - Write the sector area formula.

In this problem: A sector is the fraction of the circle that the angle cuts out, which gives \(A = \frac{1}{2} \times r^{2} \times \theta_{rad}\).

\(A = \frac{1}{2} \times r^{2} \times \theta_{rad}\)

Step 4

Step 4 - Substitute the radius and the angle in radians.

In this problem: Replacing \(r\) with \(3\) and \(\theta_{rad}\) with \(2.62\) gives \(A = \frac{1}{2} \times 3^{2} \times 2.62\).

\(A = \frac{1}{2} \times 3^{2} \times 2.62\)

Step 5

Step 5 - Square the radius.

In this problem: Squaring \(3\) gives \(9\).

\(A = \frac{1}{2} \times 9 \times 2.62\)

Step 6

Step 6 - Multiply to get the area.

In this problem: Multiplying \(9\) by \(2.62\) and halving gives \(11.78\).

\(A = \frac{1}{2} \times 9 \times 2.62 \approx 11.78\)

Final Answer

\(A \approx 11.78\)

Example Problems

Click one to load it into the box and solve it.