What is a quarter circle?
A quarter circle is exactly one fourth of a full circle — the shape you get by cutting a circle along two perpendicular radii. It is bounded by a curved arc and two straight edges (each equal to the radius). This calculator finds three key measurements of a quarter circle: its area, its arc length, and its full perimeter, using only the radius.
How to use the calculator
Enter the radius r of the quarter circle in any unit (cm, m, inches, etc.) and press calculate. The results are returned in the matching square units (for area) and linear units (for arc and perimeter). All values share the same unit you started with.
The formulas explained
Because a quarter circle is a quarter of a full circle, its area is the full-circle area divided by four: $$A = \frac{\pi r^{2}}{4}$$ Likewise, the curved arc is a quarter of the circumference (\(2\pi r\)), giving $$\text{arc} = \frac{\pi r}{2}$$ The perimeter is the arc plus the two straight radius edges that form the corner: $$P = \frac{\pi r}{2} + 2r$$
Worked example
Suppose the radius is 10 units. The area is \(\pi \times 10^{2} / 4 = 100\pi / 4 \approx 78.54\) square units. The arc length is \(\pi \times 10 / 2 = 5\pi \approx 15.71\) units. The perimeter is \(15.71 + 2 \times 10 = 35.71\) units.
FAQ
Is the perimeter the same as the arc length? No. The arc is only the curved part. The perimeter also includes the two straight radii, so it is always larger.
What units does it use? Whatever unit you enter for the radius. Area comes out in those units squared; arc and perimeter in the same linear units.
Can I use it for a semicircle or sector? This tool is specifically for a 90° quarter. For other angles, a general circular-sector calculator with an angle input is needed.
