How to Calculate the Area and Perimeter of a Circular Sector

A circle sector with an angle of theta, a radius of r, and an arc length of L.
A circle sector with an angle of theta, a radius of r, and an arc length of L.

A sector is a type of two-dimensional geometric shape that comes from a portion of a circle. A circular sector is bounded by two straight lines equal to the radius of the circle and a circular arc. The angle between the two radii is the angle of the sector.

Typically, the radius of the circle is denoted by r, the angle by θ, and the arc length by L. If you know any two of these values, you can compute the third missing value, as well as the area and perimeter of the sector.

Equations Relating r, θ, and L

Any of the three measurements can be computed if you know the values of the other two.

  • If r and θ are given, then L can be computed with the equation L = 2πrθ/360, where π is the mathematical constant equal to 3.14159... and θ is measured in degrees. If θ is measured in radians, the equation for L is L = rθ.
  • If r and L are given, then the degree measure of θ is given by the equation θ = 360L/(2πr). The radian measure of θ is simply L/r.
  • Lastly, if θ and L are given, then r = 360L/(2πθ), where θ is measured in degrees. If θ is measured in radians, then r = L/θ.

Always keep in mind whether θ is being measured in degrees or radians.

Equations for the Area of Circle Sector

The area of a circular wedge is equal to the area of the full circle times the ratio of θ to the total angle. The total angle is 360 degrees or 2π radians. Thus, the area formula is

Area = πr^2(θ/360) [degree formula]

Area = r^2(θ/2) [radian formula]

These formulas give the area in terms of r and θ. If you don't know r, but you do know L and θ, the equivalent circular sector area formula is

Area = (360L^2)/(4πθ) [degree formula]

Area = (L^2)/(2θ) [radian formula]

Alternatively, if you know the values of r and L but θ is unknown, you can compute the area of a circular sector with this formula instead:

Area = Lr/2

Equation for the Perimeter of a Circle Sector

The perimeter of any flat geometric figure is simply the sum of the lengths of all the boundary lines an curves. In a circular sector, there is the arc and the two radii. The sum is L + r + r = L + 2r, which gives us the simple perimeter formula

Perimeter = L + 2r

Example 1

A sector has a radius of 5 and an angle of 60 degrees (π/3 radians). Since r = 5 and θ = π/3, we have

Area = 5*5*(π/3)/2 = 25π/6

L = 5*π/3 = 5π/3

Perimeter = L + 2r
= 5π/3 + 10

Example 2

The perimeter of a sector is 53.7 cm. If the angle of the sector is 75 degrees, what is the area of the shape?

For this problem, we first need to solve for r using the perimeter formula for a sector, P = 2r + rθ, where θ is in radians. Since 75 degrees = 5π/12 radians, we have

53.7 = 2r + (5π/12)r
53.7 = (2 + 5π/12)r
53.7/(2 + 5π/12) = r
16.22848 = r

Now that we have the radius, we can use the area formula, A = (θ/2)r^2, where θ is in radians. This gives us

A = (5π/24)*16.22848^2
A = 172.37111

So the area is about 172.3 square centimeters.

Example 3

The arc length of a sector is is 2 times the radius. If the area of the sector is 174.5041 square units, what is the perimeter and angle of the sector? For this problem, we have L = 2r. Using the area equation Area = Lr/2, we have

174.5041 = 2r^2 / 2
174.5041 = r^2
13.21 = r

The perimeter is

2r + L = 4r
= 52.84.

The angle is

θ = L/r
= 2r/r
= 2 radians
≈ 114.59 degrees

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Comments 3 comments

Enzz 2 years ago

Hi, we learned about this last week and it's going to be on the test. Can you tell me if I solved this correctly? Two sectors, one with an angle of pi/4 and the other with angle of 2pi/3. What is the ratio of their radii so that they have equal area?(Hint solve for r/R)

r^2 * pi/4 = R^2 * 2pi/3

r^2 / R^2 = (2pi/3) / (pi/4)

(r/R)^2 = 8/3

r/R = (8/3) ^ (1/2)

thank you!

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calculus-geometry 2 years ago from Germany Author

You got the right answer, good work.

hawlet 16 months ago

how can I find the perimeter of the remaining part of circle if the angle and radius of the sector is given

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