Arc Length Calculator
A 90-degree arc on a circle with radius 5 has an arc length of about 7.85. This arc length calculator finds the arc length and sector area from the radius and the central angle, in degrees or radians. Enter the radius, enter the angle, choose the angle unit, and the tool returns the arc length, the sector area, and the full circle's circumference and area for comparison. Arc length and sector area come up constantly in geometry, engineering, and design work involving curved shapes, such as sizing a curved section of trim, calculating the fabric needed for a pie-slice-shaped panel, or working out how far a point travels along a circular path. This calculator handles both common angle formats, degrees and radians, so you never need to convert by hand before getting an answer.
Quick answer
Arc length is the distance along the curved edge of a circle section, not the straight-line distance between its endpoints.
What this tells you
- •Arc length is the distance along the curved edge of a circle section, not the straight-line distance between its endpoints.
- •The formula is arc length = radius x angle in radians.
- •Sector area is half the radius squared times the angle in radians, the wedge-shaped area bounded by two radii and the arc.
- •Degrees must be converted to radians before use in either formula, since both formulas are defined in terms of radians.
- •A full 360-degree angle returns the full circumference and the full circle area, since the arc becomes the entire circle.
How to Use
- 1Enter the radius of the circle. Any positive number works, including decimals.
- 2Enter the central angle, the angle at the circle's center that defines the arc.
- 3Choose whether the angle is in degrees or radians.
- 4Click Calculate to get the arc length, sector area, and full circle values for comparison.
- 5Use the full circle values shown alongside the result to sanity-check that your arc is a reasonable fraction of the whole circle.
How It Works
Formula
Arc length = r x theta (theta in radians). Sector area = 0.5 x r squared x theta.Arc length is the product of the radius and the central angle measured in radians. If the angle is in degrees, it is converted to radians by multiplying by pi and dividing by 180. The sector area is half of the radius squared times the angle in radians. For a 90-degree angle on a radius-5 circle, that means converting 90 degrees to pi over 2 radians, giving an arc length of 5 times pi over 2, or about 7.854, and a sector area of 0.5 times 25 times pi over 2, or about 19.635.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
90-degree arc with radius 5
Convert 90 degrees to pi/2 radians. Arc length = 5 x pi/2 = 7.854. Sector area = 0.5 x 25 x pi/2 = 19.635.
1 radian arc with radius 10
With a radius of 10 and an angle of exactly 1 radian, the arc length is simply 10. The sector area is 0.5 x 100 x 1 = 50.
60-degree arc with radius 8
60 degrees converts to pi/3 radians. Arc length = 8 x pi/3 = 8.3776. Sector area = 0.5 x 64 x pi/3 = 33.5103, a common calculation for a hexagonal-style wedge cut.
A semicircle with radius 3
180 degrees converts to pi radians, exactly half the full circle. Arc length = 3 x pi = 9.4248, which is half the circle's circumference of 6 x pi.
45-degree arc with radius 12
45 degrees converts to pi/4 radians. Arc length = 12 x pi/4 = 9.4248. Notice this matches the semicircle example above by coincidence, since different radius and angle combinations can produce the same arc length.
Arc Length for Common Angles at Radius 1
Arc length and sector area for a unit circle (radius 1).
| Angle (deg) | Arc Length | Sector Area |
|---|---|---|
| 30 | 0.5236 | 0.2618 |
| 45 | 0.7854 | 0.3927 |
| 60 | 1.0472 | 0.5236 |
| 90 | 1.5708 | 0.7854 |
| 180 | 3.1416 | 1.5708 |
| 360 | 6.2832 | 3.1416 |
Why radians are the natural unit for arc length
A radian is defined so that an angle of exactly 1 radian on a circle of radius 1 produces an arc length of exactly 1. That built-in relationship is what makes the arc length formula so simple in radians: just multiply radius by angle, with no extra conversion factor needed.
Degrees do not have that same direct relationship, since a full circle is 360 degrees for historical reasons rather than a value tied to the circle's own geometry. That is why any arc length or sector area formula written in degrees needs an extra conversion step, multiplying by pi and dividing by 180, before the math works.
Common mistakes
- Using degrees directly in the formula without converting to radians. Multiply degrees by pi and divide by 180 first.
- Confusing arc length with chord length. Arc length measures along the curve, while chord length is the straight-line distance between the two endpoints.
- Entering the diameter instead of the radius. The formula uses the radius, so divide the diameter by 2 first.
- Forgetting that sector area and arc length are different quantities in different units. Arc length is a linear measurement, while sector area is a squared measurement.
- Assuming a larger angle always means a larger arc length regardless of radius. A large angle on a small circle can still produce a shorter arc than a small angle on a large circle.
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