Circle Area, Perimeter, and Arc Length Calculator

Three related measurements from one radius

A circle with radius 10 has an area of 314.16 (pi x r²) and a circumference of 62.83 (2 x pi x r). A 90-degree arc of that same circle -- exactly a quarter of the full circle -- measures 15.71 in length.

The arc length formula (2 x pi x r x (degrees/360)) is simply the full circumference scaled down by whatever fraction of the full 360 degrees the arc represents -- at exactly 90 degrees (a quarter turn), the arc length is exactly one-quarter of the full circumference: 62.83 / 4 = 15.71, confirming the two calculations agree.

Area and circumference both scale differently with radius -- circumference is directly proportional to r (double the radius, double the circumference), while area is proportional to r² (double the radius, and area quadruples), which is why area grows so much faster than perimeter as a circle gets larger.

Using degrees, not radians, for arc angles

Arc = 2 x pi x r x (degrees/360). Full circle is 360 degrees. This calculator takes the arc angle in degrees (the more commonly used everyday unit) rather than radians, which is the unit more common in advanced math and physics contexts -- if you're working from a radian measurement, convert it to degrees first (multiply by 180/pi) before entering it here.

A full 360-degree 'arc' is mathematically identical to the full circumference, and a 0-degree arc has zero length -- the formula scales smoothly and correctly across the entire 0-360 range, with 90 degrees (as in this example) landing exactly at the quarter-circle mark.

This calculates arc length (a linear distance along the curve), not the area of the circular sector that arc encloses -- those are two different measurements of the same wedge-shaped piece of the circle, and this tool reports only the curved-edge length, not the pie-slice area.

Related geometry tools

For extending a circular cross-section into a 3D solid (a cylinder or cone), the Volume of 3D Shapes Calculator uses this same radius concept.

For a right-triangle-based geometry problem instead of a circle, the Pythagorean Triangle Solver is the related tool.

Frequently Asked Questions (FAQ)