Sector Area Calculator

Calculate the area of a sector using the central angle and radius below and learn the formula and steps to solve it below.

diagram of a sector showing the radius, central angle, chord length, and arc length

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Sector Area

Arc Length (s)

Chord Length (a)

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How to Calculate Sector Area

A sector is a pie-slice-shaped portion of a circle surrounded by two radii edges and bounded by the circle’s outer arc. A defining characteristic of a sector is its central angle, denoted θ (Greek letter theta).

If the central angle is greater than 180°, then it is a major sector, and if the central angle is less than 180°, it is a minor sector. If the central angle is exactly equal to 180° then it is a semi-circle.

graphic comparing a major and minor sector, and showing the radius, central angle, chord, and arc for a sector

Another characteristic of a sector is the chord, which is the line that connects the two points where the radii intersect the arc. The chord subdivides a sector into a triangle and a segment.

To calculate the area of a sector, a simple formula can be used.

Sector Area Formula

The area of a sector can be found using the formula:

sector area = 1 / 2r²θ

Thus, a sector’s area is equal to the radius r squared times the central angle θ in radians, divided by 2. If you know the diameter of the circle, you can find the radius by dividing the diameter in half.

graphic showing the area formulas for a sector using a central angle in degrees or radians

For example, find the area of a sector with a radius of 12 and a central angle of 1.5 radians.

area = 1 / 2 × 12² × 1.5
area = 1 / 2 × 144 × 1.5
area = 1 / 2 × 216
area = 108

Thus, the area of the sector is 108.

Sector Area Using Degrees

If you know the central angle in degrees, then the formula to find the area of a sector is a little different:

sector area = θ / 360 × πr²

The area is equal to the central angle θ in degrees divided by 360, times pi times the radius squared.

This formula is derived from the formula to find the area of a circle, which states that the area of a circle is equal to πr². Since the central angle of a circle is equal to 360° and a sector is a portion of the circle, the first part finds the portion of the full circle area represented by the sector.

For example, find the area of a sector with a radius of 7 and a central angle of 40 degrees.

area = 40 / 360 × π× 7²
area = 1 / 9 × π× 49
area = 17.104

The area is equal to 17.104.

Semi-Circle Area Formula

A semi-circle is a special type of sector equal to exactly half of a circle. The central angle of a semi-circle is 180°.

The formula to find the area of a semi-circle is:

semi-circle area = πr² / 2

The area is equal to pi times the radius squared, divided by 2.

Quadrant Area Formula

A quadrant is another special type of sector equal to exactly one-quarter of a circle. The central angle of a quadrant is 45°.

The formula to find the area of a quadrant is:

quadrant area = πr² / 4

The area is equal to pi times the radius squared, divided by 4.

You might also find our arc length calculator useful for solving the arc length of a sector.