Sector Area Calculator — Step-by-Step Solutions

The Sector Area Calculator computes the area of a circular sector from its radius and central angle. It supports both degrees and radians and provides four modes: calculate sector area, find arc length, determine the angle from a known area and radius, or find the radius from a known area and angle.

Each calculation includes step-by-step working and an SVG diagram that highlights the sector within a circle. The calculator also shows the sector perimeter (arc length plus two radii) for each computation.

Angle unit:

Radius (r)

Angle (degrees)

Your calculations will appear here

How to Use the Sector Area Calculator

Choose a mode: Sector Area, Arc Length, Find Angle, or Find Radius.
For Sector Area and Arc Length modes, select degrees or radians, then enter the radius and angle.
For Find Angle mode, enter the sector area and radius. The calculator determines the central angle.
For Find Radius mode, enter the sector area and angle. The calculator determines the radius.
View the step-by-step working and SVG sector diagram below the results.

Formula and Theory

A sector is the region of a circle enclosed by two radii and an arc. The area of a sector is a fraction of the full circle area, proportional to the central angle.

In degrees, the sector area formula is A = (theta / 360) times pi times r squared. In radians, the equivalent formula is A = (1/2) times r squared times theta, which is often simpler to work with.

The arc length of a sector is the portion of the circumference subtended by the central angle. The formula is s = r times theta (where theta is in radians). In degrees, this becomes s = (theta / 360) times 2 pi r.

The perimeter of a sector is the arc length plus two radii: P = s + 2r. This is useful for calculating the boundary length of sector-shaped regions such as pizza slices or pie charts.

Sectors appear frequently in engineering (cam profiles, gear teeth), architecture (arched windows), and data visualisation (pie charts). The radian-based formula is preferred in calculus and physics because it simplifies differentiation and integration.

Worked Examples

Example 1: Sector area from degrees

Problem: Find the area of a sector with radius 10 and central angle 60 degrees.

Solution: A = (60/360) times pi times 10 squared = (1/6) times pi times 100 = 100pi/6.

Answer: 52.3599

Example 2: Arc length from radians

Problem: Find the arc length for a sector with radius 8 and angle pi/3 radians.

Solution: s = r times theta = 8 times pi/3.

Answer: 8.3776

Example 3: Find the angle from area and radius

Problem: A sector has area 25 and radius 5. What is the central angle?

Solution: theta = 2A / r squared = 2 times 25 / 25 = 2 radians.

Answer: 2 radians (114.592 degrees)

Example 4: Find the radius from area and angle

Problem: A sector has area 50 and central angle 90 degrees.

Solution: theta in radians = pi/2. r = sqrt(2A / theta) = sqrt(2 times 50 / (pi/2)) = sqrt(200 / pi * 2).

Answer: 7.9789

Frequently Asked Questions

What is the difference between a sector and a segment?

A sector is enclosed by two radii and an arc (like a pizza slice). A segment is enclosed by a chord and an arc (the region between a chord and the arc it subtends).

When should I use degrees versus radians?

Use degrees for everyday measurements and radians for calculus, physics, and engineering. The radian formula A = (1/2)r squared theta is simpler and avoids the 360 divisor.

Can the angle be greater than 360 degrees?

For a single sector, the angle must be between 0 and 360 degrees (or 0 to 2pi radians). An angle of 360 degrees gives the full circle area.

How is sector area related to pie charts?

Each slice of a pie chart is a sector. The central angle of each slice is proportional to the data value it represents relative to the total.

Related Trigonometry Calculators

triangle

Triangle Calculator

Calculate triangle sides, angles, area, and perimeter. Supports SSS, SAS, ASA, AAS configurations. Free online triangle calculator.

sine-wave

Unit Circle Calculator

Interactive unit circle with exact trigonometric values. Find sin, cos, tan for any angle, reference angles, and degree-radian conversions. Free, instant results.

percentage