An arc is any portion of the circumference of a circle. Knowing the radius and the central angle that subtends it is enough to calculate the exact length of that curved path, and this tool lets you solve in any direction: find the arc length from radius and angle, find the radius from arc length and angle, or find the angle from arc length and radius. It also shows the chord (the straight-line shortcut between the two arc endpoints), the sector area (the pie-slice of the circle bounded by the arc and the two radii), and a live SVG diagram that updates as you type.
How it works
The central relationship is simply:
L = r x θ
where L is the arc length, r is the radius, and θ is the central angle
in radians. Because angles are usually given in degrees, the first step is
always converting: θ (rad) = θ (deg) x π ÷ 180.
Rearranging the formula gives the other two solve modes:
| Solve for | Formula |
|---|---|
| Arc length | L = r x θ |
| Radius | r = L ÷ θ |
| Central angle | θ = L ÷ r |
The tool also derives two bonus quantities from r and θ:
- Chord length =
2 x r x sin(θ ÷ 2)— the straight line joining the arc’s endpoints. At exactly 180° it equals the diameter (2r); for very small angles it is almost identical to the arc length. - Sector area =
0.5 x r² x θ— the area of the “pie slice” bounded by the two radii and the arc. At θ = 2π (a full circle) this reduces to the familiarπ x r².
Worked example
A clock’s minute hand is 12 cm long. It sweeps through 90° in 15 minutes. How far does the tip travel?
- Convert the angle: θ = 90 x π ÷ 180 = π/2 ≈ 1.5708 rad
- Apply the formula: L = 12 x 1.5708 = ≈ 18.85 cm
Bonus figures for the same inputs:
| Quantity | Value |
|---|---|
| Arc length | ≈ 18.850 cm |
| Chord length | ≈ 16.971 cm |
| Sector area | ≈ 113.097 cm² |
Now suppose you know the arc is 25 cm long and the angle is 60°. What is the radius?
- Convert: θ = 60 x π ÷ 180 ≈ 1.0472 rad
- r = L ÷ θ = 25 ÷ 1.0472 ≈ 23.873 cm
Both of these are computed instantly in the calculator above — the step-by-step working panel shows every intermediate value so you can follow along or copy the steps into your homework.
Formula note
The formula L = r x θ is exact, not an approximation. It follows directly from
the definition of the radian: one radian is the angle for which the arc length
equals the radius. So an angle of θ radians corresponds to an arc that is θ times
the radius. There is no Taylor-series truncation or small-angle approximation here;
the result is precise for any angle from just above 0 to a full 360° circle.
For a full revolution (θ = 2π), L = r x 2π = the full circumference, which is the
familiar formula C = 2πr — confirming the formula is consistent.
All arithmetic runs in your browser with JavaScript’s 64-bit floating-point numbers, and results are rounded to six significant decimal places for display.