Quarter Circle Calculator

Background.

A quarter circle turns up wherever a curve meets a right-angled corner: a flowerbed tucked into the angle of two fences, the sweep of a door across the floor, a rounded countertop corner, a corner shower tray, the cross-section of quarter-round trim. The shape is always the same, a 90° slice of a circle bounded by a curved arc and two straight edges that meet at the centre at a right angle, each as long as the radius.

Its area is a quarter of the full circle's πr², so πr²/4. A handy way to picture it is that the quarter circle fills π/4, about 78.5%, of the r-by-r square it sits in. A corner bed with a 2.5 m radius covers about 4.91 m², while the square of lawn it is cut from measures 6.25 m².

Perimeter is where estimates go wrong. The curved edge alone is a quarter of the circumference, πr/2, but the full boundary also includes both straight radii, adding 2r. For any radius the straight sides are the longer part, 2r against about 1.571r, so counting only the curve leaves you short by more than half the true perimeter. For the 2.5 m bed the arc is about 3.93 m and the whole boundary about 8.93 m; if the bed's straight sides run along fences, the 3.93 m of arc is all the edging you need.

Because area grows with the square of the radius and perimeter only in proportion to it, doubling the radius quadruples the area but merely doubles the boundary. The page works in whatever length unit you enter, and areas come out in that unit squared. It handles exactly 90° slices; a sector of any other angle needs its own calculation.

What is quarter circle calculator?

A quarter circle, or quadrant, is the region of a circle between two radii at right angles to each other. It has three edges, an arc one quarter of the circumference long and two straight radii, and one right-angled corner at the circle's centre. Its area is one quarter of the whole circle's, and its perimeter is the arc plus both radii, not the arc alone. A single measurement defines it completely: the radius, the distance from the corner to any point on the curve.

How to use this calculator.

1. 01Measure the radius from the right-angled corner to the curved edge. On a real object, check two or three points along the curve; if the distances differ, the shape is not a true quarter circle.
2. 02Enter the radius in any length unit. The area comes back in that unit squared and the perimeter in the same unit.
3. 03Use the area for material that covers the surface, such as turf, paving, paint or tile, and the perimeter for anything that runs round the edge. If the straight sides sit against walls, subtract twice the radius and keep only the arc.
4. 04If you only know a diameter, halve it before entering it.

The formula.

Area A = πr²/4 and perimeter P = πr/2 + 2r, where r is the radius. The area takes the full circle's πr² and keeps one quarter, because a right angle is a quarter of a full turn. In the perimeter, πr/2 is the arc, a quarter of the circumference 2πr, and 2r is the pair of straight edges running from the corner to the ends of the arc. Since π/2 ≈ 1.571 is less than 2, the straight edges always make up more of the perimeter than the curve does, about 56% of it. Area scales with r², so a radius 10% larger gives 21% more area but only 10% more perimeter. π is carried at full precision rather than as 3.14, so the results are not limited by a rounded constant.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.