What Is a Quarter Circle?
A quarter circle is exactly one-fourth of a full circle — the shape you get when you cut a circle with two perpendicular radii, like one slice of a pie divided into four equal pieces. Because it represents 90° out of 360°, its area is simply one quarter of the whole circle's area. This calculator finds that area instantly from the radius, and also gives you the length of the curved edge and the full perimeter of the shape.
How to Use This Calculator
Enter the radius (\(r\)) of the quarter circle in whatever unit you like — centimeters, inches, meters, etc. — and the tool returns the area in those units squared. The radius is the straight distance from the corner (the center of the original circle) to the curved edge. All three results scale automatically with your chosen unit.
The Formula Explained
A full circle has area \(\pi r^{2}\). Since a quarter circle is one-fourth of that, the formula is:
$$A = \frac{\pi r^{2}}{4}$$The curved edge (quarter of the circumference) has length \(L = \dfrac{\pi r}{2}\), and the complete perimeter — the arc plus the two straight radii — is \(P = \dfrac{\pi r}{2} + 2r\).
Worked Example
Suppose the radius is 10 units. Then $$A = \frac{\pi \times 10^{2}}{4} = \frac{\pi \times 100}{4} = 25\pi \approx 78.54 \text{ square units}.$$ The arc length is \(\pi \times 10 / 2 = 5\pi \approx 15.71\) units, and the perimeter is \(15.71 + 20 = 35.71\) units.
FAQ
Is a quarter circle area exactly one-fourth of a circle? Yes. Because the angle is 90° (a quarter of 360°), the area is exactly \(\pi r^{2}/4\).
What units does the result use? If you enter the radius in centimeters, the area is in square centimeters. The unit always matches the square of whatever you input.
Does the perimeter include the straight sides? Yes — the perimeter reported here is the curved arc plus both straight radii, the complete outline of the quarter-circle region.
