Perimeter of Sector Calculator
Enter the radius and the central angle, in degrees or radians, to find the perimeter of a sector.
Updated
Perimeter of Sector Calculator
Our Perimeter of Sector Calculator adds the two straight radii of a sector to its curved arc to give the full distance around the slice. Enter the radius and the central angle in degrees or radians, and it shows the working, the exact value in terms of π, the arc length and the area.
Quick answer
The perimeter of a sector is P = 2r + s, where the arc is s = (θ ÷ 360) × 2πr: r = 7 cm, θ = 90° gives 14 + 7π/2 ≈ 25 cm.
- Angle in radians: s = rθ, so P = r(2 + θ); r = 10 and θ = 2 rad give 40
- Half circle (180°): P = πr + 2r, the same as a semicircle
- Angle from the perimeter: θ = P ÷ r − 2 radians
- Arc only: the curved edge alone is s; the perimeter adds both radii
What Is the Perimeter of a Sector?
A sector is the slice of a circle between two radii and the arc that joins them, like a slice of pizza. Its perimeter is the total length of its edge: two straight radii plus the curved arc. It is measured in ordinary length units such as cm, m or ft.
Perimeter of a Sector Formula
P = 2r + s
P = r(2 + θ) (θ in rad)
Worked example for a radius of 7 cm and a central angle of 90°:
- Find the share of the circle. 90 ÷ 360 = 1/4.
- Find the arc. 1/4 × 2π × 7 = 7π/2 ≈ 11 cm.
- Find the two radii. 2 × 7 cm = 14 cm.
- Add them. 14 + 11 ≈ 25 cm (more precisely 24.9956 cm).
Leave the answer as 14 + 7π/2 cm for an exact value. For the arc alone, use the Area of Sector Calculator, which also lists it.
Sector Perimeter Examples
| r, θ | Arc (s) | Perimeter |
|---|---|---|
| 5 cm, 45° | ≈ 3.93 cm | ≈ 13.93 cm |
| 10 m, 60° | ≈ 10.47 m | ≈ 30.47 m |
| 7 cm, 90° | ≈ 11 cm | ≈ 25 cm |
| 10 m, 120° | ≈ 20.94 m | ≈ 40.94 m |
| 15 in, 180° | ≈ 47.12 in | ≈ 77.12 in |
| 12 ft, 270° | ≈ 56.55 ft | ≈ 80.55 ft |
The two radii stay the same whatever the angle, so only the arc grows as the slice gets wider. At 360° the arc is the whole circumference and the perimeter is 2πr + 2r.
Perimeter With the Angle in Radians
In radians the arc is simply the radius times the angle, s = rθ, so the perimeter is P = 2r + rθ = r(2 + θ). No π is needed unless the angle itself is a multiple of π.
P = 10 × (2 + 2) = 40
r = 6, θ = 1.5 rad
P = 6 × 3.5 = 21
You can type angles such as π/3 or 2pi/3 straight into the calculator when rad is selected. A radius of 8 in and θ = π rad gives 16 + 8π ≈ 41.13 in.
Angle From the Perimeter
Turn the formula around to find the central angle when you know the perimeter and the radius. Take away the two radii to get the arc, then divide by r.
(30 − 20) ÷ 10 = 1 rad
1 rad ≈ 57.2958°
Common Mistakes
- Leaving out the radii. s is only the arc; the perimeter is 2r + s.
- Mixing up angle units. s = rθ needs θ in radians; with degrees use (θ ÷ 360) × 2πr.
- Using the diameter as the radius. Both the arc and the radii come out twice too big.
- Square units. A perimeter is a length: 25 cm, not cm².
Key Terms
- Sector
- A slice of a circle between two radii.
- Radius
- Centre to arc, r.
- Angle
- The central angle θ at the centre.
- Arc length
- The curved edge, s.
- Perimeter
- Two radii plus the arc, 2r + s.
How to Use the Perimeter of Sector Calculator
- Enter the radius. Type the radius of the circle, such as 7, and pick a unit if you want one (mm, cm, m, km, in, ft, yd or mi). Decimals and fractions such as 3/4 work.
- Enter the central angle. Type the angle, such as 90, and choose deg or rad. With rad you can also type multiples of π such as π/3 or 2pi/3.
- Read and copy the result. The perimeter appears as you type, with a drawing, the working, the exact value in terms of π, the arc length, the two radii, the angle in the other unit and the sector area. Press Copy to copy it, or Clear or Esc to start again.
Example: a quarter-circle flower bed has a 3 m radius. Enter 3, pick m and type 90 with deg. The edging needed is about 10.712389 m, of which the curved part is about 4.712389 m.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
How do you find the perimeter of a sector?
Add the two radii to the arc length: P = 2r + s. A sector with a 7 cm radius and a 90° angle has an arc of 7π/2 ≈ 11 cm, so its perimeter is 14 + 7π/2 ≈ 25 cm.
What is the formula for the perimeter of a sector?
P = 2r + (θ ÷ 360) × 2πr with the angle θ in degrees, or P = 2r + rθ = r(2 + θ) with θ in radians.
How do you find the perimeter of a sector in radians?
Multiply the radius by the angle to get the arc, then add both radii. With r = 10 and θ = 2 rad the arc is 20 and the perimeter is 40.
Is the perimeter of a sector the same as the arc length?
No. The arc length is only the curved edge. The perimeter also includes the two straight radii, so it is always 2r longer than the arc.
How do you find the angle of a sector from its perimeter?
Subtract 2r from the perimeter to get the arc, then divide by r for the angle in radians: θ = (P − 2r) ÷ r. A perimeter of 30 with r = 10 gives 1 rad, about 57.2958°.
What is the perimeter of a semicircle sector?
A 180° sector is a semicircle, so its perimeter is πr + 2r. For a radius of 15 in that is about 77.12 in.
Is the Perimeter of Sector Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Perimeter of Sector Calculator to find the distance around any slice of a circle from its radius and angle, in degrees or radians.