Arc Length Calculator

This Arc Length Calculator helps you find the length of an arc when the radius and central angle are known. It first converts the angle from degrees to radians, then uses the formula s = rθ, where r is the radius and θ is the angle in radians. Multiply the radius by the angle in radians to get the arc length. It is a simple way to check answers, understand arc length 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 arc length formula and calculate.

Formulas:

Example 1:

Step 1 - Identify what is given.

In this problem: The given values are \(r = 5\) and \(\theta = 60^{\circ}\).

Step 2 - Convert the angle from degrees to radians.

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

Step 3 - Use the arc length formula and calculate.

In this problem: Use \(L = r \times \theta_{rad}\): \(L = 5 \times 1.05 \approx 5.24\).

Final answer:

Example 2:

Step 1 - Identify what is given.

In this problem: The given values are \(r = 10\) and \(\theta = 90^{\circ}\).

Step 2 - Convert the angle from degrees to radians.

In this problem: Convert degrees to radians using \(\theta_{rad} = \frac{\theta \times \pi}{180}\): \(\theta_{rad} = \frac{90^{\circ} \times \pi}{180} \approx 1.57\).

Step 3 - Use the arc length formula and calculate.

In this problem: Use \(L = r \times \theta_{rad}\): \(L = 10 \times 1.57 \approx 15.71\).

Final answer:

See Example 1 Hide Example 1

Problem

\(r = 5,\; \theta = 60^{\circ}\)

Approach

Step-by-step method

1. Identify what is given.
2. Convert the angle from degrees to radians.
3. Use the arc length formula and calculate.

Step 1

Step 1 - Identify what is given.

In this problem: The given values are \(r = 5\) and \(\theta = 60^{\circ}\).

\(r = 5,\; \theta = 60^{\circ}\)

Step 2

Step 2 - Convert the angle from degrees to radians.

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

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

Step 3

Step 3 - Use the arc length formula and calculate.

In this problem: Use \(L = r \times \theta_{rad}\): \(L = 5 \times 1.05 \approx 5.24\).

\(L = 5 \times 1.05 \approx 5.24\)

Final Answer

\(r = 5,\; \theta = 60^{\circ} = L \approx 5.24\)

See Example 2 Hide Example 2

Problem

\(r = 10,\; \theta = 90^{\circ}\)

Approach

Step-by-step method

1. Identify what is given.
2. Convert the angle from degrees to radians.
3. Use the arc length formula and calculate.

Step 1

Step 1 - Identify what is given.

In this problem: The given values are \(r = 10\) and \(\theta = 90^{\circ}\).

\(r = 10,\; \theta = 90^{\circ}\)

Step 2

Step 2 - Convert the angle from degrees to radians.

In this problem: Convert degrees to radians using \(\theta_{rad} = \frac{\theta \times \pi}{180}\): \(\theta_{rad} = \frac{90^{\circ} \times \pi}{180} \approx 1.57\).

\(\theta_{rad} = \frac{90^{\circ} \times \pi}{180} \approx 1.57\)

Step 3

Step 3 - Use the arc length formula and calculate.

In this problem: Use \(L = r \times \theta_{rad}\): \(L = 10 \times 1.57 \approx 15.71\).

\(L = 10 \times 1.57 \approx 15.71\)

Final Answer

\(r = 10,\; \theta = 90^{\circ} = L \approx 15.71\)