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quarter circle — Definition, Formula & Examples

A quarter circle is exactly one-fourth of a full circle, formed by two radii that meet at a 90° angle and the arc between them. It looks like a slice of pie or the curved corner of a square.

A quarter circle (or quadrant) is a sector of a circle with a central angle of 90°, bounded by two perpendicular radii and the intercepted arc of measure π2\frac{\pi}{2} radians.

Key Formula

A=14πr2P=2r+πr2A = \frac{1}{4}\pi r^2 \qquad P = 2r + \frac{\pi r}{2}
Where:
  • AA = Area of the quarter circle
  • PP = Perimeter (total boundary length) of the quarter circle
  • rr = Radius of the full circle

How It Works

To find measurements of a quarter circle, take the corresponding full-circle formula and divide by 4. The area is 14πr2\frac{1}{4}\pi r^2. The arc length (the curved part) is 14\frac{1}{4} of the circumference, or πr2\frac{\pi r}{2}. If you need the full perimeter — the total distance around the quarter circle — add the two straight edges (each of length rr) to the arc length, giving 2r+πr22r + \frac{\pi r}{2}.

Worked Example

Problem: Find the area and perimeter of a quarter circle with radius 10 cm.
Area: Use the quarter-circle area formula.
A=14π(10)2=100π4=25π≈78.54 cm2A = \frac{1}{4}\pi (10)^2 = \frac{100\pi}{4} = 25\pi \approx 78.54 \text{ cm}^2
Arc length: The curved edge is one-fourth of the full circumference.
Arc=π(10)2=5π≈15.71 cm\text{Arc} = \frac{\pi(10)}{2} = 5\pi \approx 15.71 \text{ cm}
Perimeter: Add the two straight sides (each 10 cm) to the arc length.
P=2(10)+5π=20+5π≈35.71 cmP = 2(10) + 5\pi = 20 + 5\pi \approx 35.71 \text{ cm}
Answer: The area is 25π≈78.5425\pi \approx 78.54 cm² and the perimeter is 20+5π≈35.7120 + 5\pi \approx 35.71 cm.

Why It Matters

Quarter circles appear in architecture (rounded corners on windows and walls), in composite-area problems on standardized tests, and whenever you need to calculate the area of a rounded corner cut from a rectangle.

Common Mistakes

Mistake: Giving only the arc length when asked for the perimeter.
Correction: The perimeter of a quarter circle includes three edges: the curved arc plus two straight radii. Always add 2r2r to the arc length.

Related Terms

  • Circle — The full shape a quarter circle comes from
  • Arc of a Circle — The curved edge of a quarter circle
  • Central Angle — A quarter circle has a 90° central angle
  • Circumference — Quarter-circle arc is one-fourth of this
  • Diameter — Equals twice the radius used in formulas