Sector Area Calculator

This Sector Area Calculator helps you find the area of a sector when the radius and central angle are known. It first converts the angle from degrees to radians, then uses the formula A = ½r²θ, where r is the radius and θ is the angle in radians. Square the radius, multiply by the angle in radians, and then multiply by ½ to get the sector area. It is a simple way to check answers, understand sector area formulas, and practise basic trigonometry step by step.

Step-by-step method

1. Identify what is given.
2. Convert the angle from degrees to radians.
3. Use the sector area formula.
4. Substitute and calculate.

Formulas:

Example 1:

Step 1 - Identify what is given.

In this problem: The given values are \(r = 4\) and \(\theta = 60^{\circ}\).

Step 2 - Convert the angle from degrees to radians.

In this problem: Convert degrees to radians using \(\theta_{rad} = \theta^{\circ} \times \frac{\pi}{180}\): \(\theta_{rad} = 60^{\circ} \times \frac{\pi}{180} \approx 1.05\).

Step 3 - Use the sector area formula.

In this problem: Use the sector area formula: \(A = \frac{1}{2} \times r^{2} \times \theta_{rad}\).

Step 4 - Substitute and calculate.

In this problem: Substitute and calculate: \(A = \frac{1}{2} \times 4^{2} \times 1.05 \approx 8.38\).

Final answer:

Example 2:

Step 1 - Identify what is given.

In this problem: The given values are \(r = 5.5\) and \(\theta = 120^{\circ}\).

Step 2 - Convert the angle from degrees to radians.

In this problem: Convert degrees to radians using \(\theta_{rad} = \theta^{\circ} \times \frac{\pi}{180}\): \(\theta_{rad} = 120^{\circ} \times \frac{\pi}{180} \approx 2.09\).

Step 3 - Use the sector area formula.

In this problem: Use the sector area formula: \(A = \frac{1}{2} \times r^{2} \times \theta_{rad}\).

Step 4 - Substitute and calculate.

In this problem: Substitute and calculate: \(A = \frac{1}{2} \times 5.5^{2} \times 2.09 \approx 31.68\).

Final answer:

See Example 1 Hide Example 1

Problem

\(r = 4,\; \theta = 60^{\circ}\)

Approach

Step-by-step method

1. Identify what is given.
2. Convert the angle from degrees to radians.
3. Use the sector area formula.
4. Substitute and calculate.

Step 1

Step 1 - Identify what is given.

In this problem: The given values are \(r = 4\) and \(\theta = 60^{\circ}\).

\(r = 4,\; \theta = 60^{\circ}\)

Step 2

Step 2 - Convert the angle from degrees to radians.

In this problem: Convert degrees to radians using \(\theta_{rad} = \theta^{\circ} \times \frac{\pi}{180}\): \(\theta_{rad} = 60^{\circ} \times \frac{\pi}{180} \approx 1.05\).

\(\theta_{rad} = 60^{\circ} \times \frac{\pi}{180} \approx 1.05\)

Step 3

Step 3 - Use the sector area formula.

In this problem: Use the sector area formula: \(A = \frac{1}{2} \times r^{2} \times \theta_{rad}\).

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

Step 4

Step 4 - Substitute and calculate.

In this problem: Substitute and calculate: \(A = \frac{1}{2} \times 4^{2} \times 1.05 \approx 8.38\).

\(A = \frac{1}{2} \times 4^{2} \times 1.05 \approx 8.38\)

Final Answer

\(A \approx 8.38\)

See Example 2 Hide Example 2

Problem

\(r = 5.5,\; \theta = 120^{\circ}\)

Approach

Step-by-step method

1. Identify what is given.
2. Convert the angle from degrees to radians.
3. Use the sector area formula.
4. Substitute and calculate.

Step 1

Step 1 - Identify what is given.

In this problem: The given values are \(r = 5.5\) and \(\theta = 120^{\circ}\).

\(r = 5.5,\; \theta = 120^{\circ}\)

Step 2

Step 2 - Convert the angle from degrees to radians.

In this problem: Convert degrees to radians using \(\theta_{rad} = \theta^{\circ} \times \frac{\pi}{180}\): \(\theta_{rad} = 120^{\circ} \times \frac{\pi}{180} \approx 2.09\).

\(\theta_{rad} = 120^{\circ} \times \frac{\pi}{180} \approx 2.09\)

Step 3

Step 3 - Use the sector area formula.

In this problem: Use the sector area formula: \(A = \frac{1}{2} \times r^{2} \times \theta_{rad}\).

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

Step 4

Step 4 - Substitute and calculate.

In this problem: Substitute and calculate: \(A = \frac{1}{2} \times 5.5^{2} \times 2.09 \approx 31.68\).

\(A = \frac{1}{2} \times 5.5^{2} \times 2.09 \approx 31.68\)

Final Answer

\(A \approx 31.68\)