Quarter Circle Calculator

Enter a quarter circle's radius to get its area (A = πr²/4), arc length (L = πr/2), perimeter (P = 2r + πr/2), and chord length (c = r√2).

Quick Facts

Area formula
A = πr² / 4
One fourth of a full circle's area.
Arc length
L = πr / 2
One fourth of the full circumference 2πr.
Chord length
c = r × √2 ≈ 1.4142r
Straight distance between the two radii endpoints, since they meet at 90°.

Your Results

Calculated
Area
-
A = πr² / 4
Arc Length
-
L = πr / 2
Perimeter
-
P = 2r + πr / 2
Chord Length
-
c = r√2

Ready

Enter a radius and unit, then press Calculate.

Formula and Method for the Quarter Circle

A quarter circle is a quarter (one fourth) of a full circle, bounded by two straight radii meeting at a 90° angle and a curved arc connecting their endpoints. Because it is exactly one fourth of a circle, its area and arc length are simply one fourth of the full circle's area (πr²) and circumference (2πr): A = πr² / 4 and L = πr / 2, where r is the radius. This calculator also derives the full perimeter of the quarter-circle shape and the straight-line chord connecting the two open ends of the radii.

How the calculation works

Enter the radius and choose the unit it is measured in. The calculator squares the radius, multiplies by π, and divides by 4 to get the area (in square units, such as ft² or m²). It multiplies the radius by π and divides by 2 to get the arc length. The perimeter adds the two straight radii to that arc: P = 2r + πr/2. The chord — the straight distance between the two endpoints of the radii — follows from the Pythagorean theorem applied to the right angle at the center: c = r√2 ≈ 1.4142r.

Common mistakes

  • Confusing arc length with perimeter: the arc length (πr/2) is only the curved edge. The full perimeter of the quarter-circle shape also includes the two straight radii: P = 2r + πr/2.
  • Diameter vs. radius: this calculator uses radius, not diameter. If you only know the diameter, divide it by 2 before entering the value — using diameter directly overstates the area by a factor of 4.
  • Mixing units: keep the radius in one consistent unit — convert inches to feet, or centimeters to meters, before entering the value.

Real-world applications

  • Landscaping and hardscaping use quarter-circle area for corner flower beds, patios, and rounded lawn sections.
  • Fabrication and sheet-metal work use the arc length and chord to lay out rounded corners and quarter-round trim pieces.
  • Architecture and interior design use quarter-circle geometry for curved staircases, rounded countertops, and fan-shaped windows.
  • Engineering drawings use the perimeter and chord to check that a rounded corner or fillet matches its specified radius.

Frequently Asked Questions

What is the formula for the area of a quarter circle?
The area of a quarter circle is one fourth of the area of a full circle: A = πr² / 4, where r is the radius. For example, a quarter circle with a 10 ft radius has an area of π × 10² / 4 ≈ 78.54 ft².
How do I find the arc length of a quarter circle?
The arc length is one fourth of the full circle's circumference: L = πr / 2 (since the full circumference is 2πr and a quarter circle spans a 90° angle). A 10 ft radius gives an arc length of π × 10 / 2 ≈ 15.71 ft.
What is the perimeter of a quarter circle?
The perimeter is the sum of the two straight radii and the curved arc: P = 2r + πr/2. For a 10 ft radius, that is 2 × 10 + π × 10 / 2 ≈ 35.71 ft.
What is the chord length of a quarter circle?
The chord is the straight-line distance connecting the two open endpoints of the radii. Since the two radii meet at a 90° angle, the Pythagorean theorem gives chord = r√2. A 10 ft radius has a chord of 10 × 1.4142 ≈ 14.14 ft.

Related calculators