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.
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.
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.
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.