Circle Sector Calculator

What Is a Circle Sector Calculator?

A Circle Sector Calculator is a geometry tool that calculates important measurements of a sector using the circle's radius and central angle.

A sector is a portion of a circle bounded by two radii and the arc between them. It looks similar to a slice of a pie, which is why sectors are often used as examples when explaining circular geometry.

By entering the radius and central angle, this calculator can determine the sector area, arc length, sector perimeter, and chord length.

The calculator also provides the corresponding measurements for the complete circle, making it easier to understand the relationship between a sector and the entire circle.

How to Use the Circle Sector Calculator

Using the calculator is simple. Enter the radius of the circle and the central angle of the sector in degrees.

1

Enter Radius

Enter the distance from the center of the circle to its edge.

2

Enter Central Angle

Enter the sector's central angle between 0 and 360 degrees.

3

View Results

Click calculate to instantly see the sector area, arc length, perimeter, and chord.

Circle Sector Formulas

Sector Area

The area of a sector is the same fraction of the full circle's area as the central angle is of 360 degrees.

A = (θ / 360) × πr²

Arc Length

Arc length represents the curved boundary of the sector.

L = (θ / 360) × 2πr

Sector Perimeter

The perimeter of a sector includes the curved arc and the two radius sides.

P = 2r + L

Chord Length

The chord is the straight-line distance connecting the two endpoints of the arc.

c = 2r × sin(θ / 2)

Circle Sector Calculation Example

Suppose a circle sector has a radius of 10 units and a central angle of 90°.

The sector represents one-quarter of the complete circle.

Its area is:

A = (90 / 360) × π × 10²

A = 25π

A ≈ 78.54 square units

Its arc length is:

L = (90 / 360) × 2π × 10

L = 5π

L ≈ 15.71 units

The sector perimeter is:

P = 2(10) + 15.71

P ≈ 35.71 units

Therefore, a 90-degree sector with a radius of 10 units has an area of approximately 78.54 square units.

Common Circle Sector Angles

The size of a sector depends on its central angle. Common angles correspond to familiar fractions of a complete circle.

90° = 1/4 of a circle

120° = 1/3 of a circle

180° = 1/2 of a circle

270° = 3/4 of a circle

360° = Full circle

The larger the central angle, the larger the sector's area and arc length will be for the same radius.

Frequently Asked Questions

What is a circle sector?

A circle sector is the region enclosed by two radii and the arc between them.

How do you calculate the area of a sector?

Multiply the full circle area by the fraction represented by the central angle: A = (θ / 360) × πr².

What is the formula for arc length?

Arc length is calculated using L = (θ / 360) × 2πr when the angle is measured in degrees.

What is the perimeter of a circle sector?

The sector perimeter is the arc length plus the two radii: P = L + 2r.

What is a central angle?

A central angle is an angle whose vertex is at the center of the circle and whose sides extend to the circle's circumference.

Can a sector angle be greater than 360 degrees?

No. A standard circle sector uses a central angle from 0 to 360 degrees.

What happens when the central angle is 180 degrees?

A 180-degree sector is exactly half of a circle, so it forms a semicircle.

What happens when the angle is 90 degrees?

A 90-degree sector represents one-quarter of a complete circle.

What is the difference between arc length and perimeter?

Arc length measures only the curved portion of the sector. Perimeter includes the arc plus both radius sides.

What is a chord in a circle sector?

A chord is the straight line connecting the two endpoints of the sector's curved arc.