Quarter Circle Perimeter Calculator
Enter the radius of a quarter circle (a quarter of a full disk) to get its full perimeter, arc length, area, chord length, external area, and centroid. Switch between metric and imperial units and use the "Solve from" selector to work backwards from perimeter, arc length, area, or diameter. Every result comes with a labeled shape diagram and a step-by-step breakdown of the math.
Formula
Worked example
For a quarter circle with radius 5 m: arc = pi x 5 / 2 = 7.8540 m; perimeter = 2 x 5 + 7.8540 = 17.8540 m; area = pi x 25 / 4 = 19.6350 m^2; chord = 5 x sqrt(2) = 7.0711 m; centroid at 4 x 5 / (3 x pi) = 2.1221 m along each axis.
What is a quarter circle?
A quarter circle (also called a quadrant) is one quarter of a full circle, formed by two straight radii meeting at a right angle and the arc between them. Its boundary has three parts: two straight edges of equal length (each one radius long) and one curved arc. The angle at the centre is always 90 degrees, so the arc spans exactly a quarter of the full circumference. Quarter circles appear everywhere, from the rounded corners of furniture and phone screens to arched bridges and fan-shaped garden beds.
Quarter circle perimeter formula
The perimeter is the total length of the boundary. Because the boundary is made of the arc and two radii, the formula is P = (pi x r / 2) + 2r, which can also be written as P = r(pi/2 + 2). For a radius of 5 m this gives P = 5 x (pi/2 + 2) = 5 x 3.5708 = 17.854 m. The arc alone is L = pi x r / 2, which equals about 1.5708 r. The arc fraction of the total perimeter is always pi / (pi + 4), roughly 44% regardless of the radius.
Area, chord, external area and centroid
The enclosed area is A = pi x r^2 / 4 - one quarter of the area of the full circle. The chord is the straight line connecting the two tips of the arc, running diagonally across the bounding square: c = r x sqrt(2). The external area is the part of the bounding square (side r) that lies outside the quarter circle: r^2 - pi x r^2 / 4. This is sometimes needed in tiling or cut-waste calculations. The centroid (centre of mass) of a uniform quarter-circle lamina is at x = y = 4r / (3 x pi) from the corner where the two radii meet.
Reverse calculations: solving for radius from other measurements
Use the "Solve from" selector to work backwards from any known value. From the arc length: r = 2L / pi. From the area: r = sqrt(4A / pi). From the perimeter: because P = r(2 + pi/2), the radius is r = P / (2 + pi/2) = P / 3.5708. From the diameter: r = d / 2. These inverses are exact, not approximations, so the calculator gives the same level of precision whichever starting point you use.
Quarter-circle properties for common radii
| Radius (r) | Perimeter | Arc length | Area | Chord |
|---|---|---|---|---|
| 1 | 3.5708 | 1.5708 | 0.7854 | 1.4142 |
| 2 | 7.1416 | 3.1416 | 3.1416 | 2.8284 |
| 3 | 10.7124 | 4.7124 | 7.0686 | 4.2426 |
| 5 | 17.8540 | 7.8540 | 19.6350 | 7.0711 |
| 10 | 35.7080 | 15.7080 | 78.5398 | 14.1421 |
| 25 | 89.2699 | 39.2699 | 490.8739 | 35.3553 |
| 50 | 178.5398 | 78.5398 | 1963.4954 | 70.7107 |
All values rounded to 4 decimal places. Units match whichever unit the radius is stated in.
Frequently asked questions
What is the perimeter of a quarter circle?
The perimeter is the arc plus the two straight radii: P = pi x r / 2 + 2r = r(pi/2 + 2). For r = 10 cm this is 10 x 3.5708 = 35.708 cm. The arc accounts for about 44% of the total perimeter; the two radii together make up the remaining 56%.
How do I find the radius from the perimeter?
Rearrange the perimeter formula P = r(2 + pi/2) to get r = P / (2 + pi/2). The divisor 2 + pi/2 is approximately 3.5708, so dividing the known perimeter by 3.5708 gives the radius. Use the "Solve from: Perimeter" option above to do this automatically.
Is the arc length half the perimeter?
No. The arc length L = pi x r / 2 is always about 44.1% of the total perimeter P = r(pi/2 + 2). The two radii together contribute the other 55.9%. The exact fraction is pi / (pi + 4), which is fixed regardless of the size of the quarter circle.
What is the chord of a quarter circle?
The chord is the straight line connecting the two endpoints of the arc - the same as the hypotenuse of the right-angled isosceles triangle formed by the two radii. Its length is c = r x sqrt(2), approximately 1.4142 r. For a radius of 5 m the chord is about 7.071 m.
How does the area of a quarter circle relate to the full circle?
The quarter-circle area A = pi x r^2 / 4 is exactly one quarter of the full circle area pi x r^2. Similarly, the arc length is exactly one quarter of the full circumference 2 x pi x r.
Where is the centroid of a quarter circle?
For a uniform flat quarter-circle plate, the centroid (centre of mass) is located at x = y = 4r / (3 x pi) from the corner where the two radii meet. For r = 5 m this is about 2.122 m along each axis. The result is the same on both axes because the shape is symmetric about the diagonal.