Quarter Circle Calculator
From a single radius, get the area, the arc length, and the full perimeter — the three numbers that describe any quarter circle (a 90° slice of a disc).
What is a quarter circle calculator?
Radius in, full quadrant out
A quarter circle calculator turns one measurement — the radius — into the numbers that describe a quarter circle (a quadrant): how much surface it covers (area), the length of its curved edge (arc length), and the distance all the way around it (perimeter). A quarter circle is exactly one-fourth of a full disc, the wedge swept by a 90° angle. Once you know the radius, every one of those numbers is fixed by the constant π (pi), which makes a single input all you need for rounded table corners, garden beds, fan-shaped patios, pie-slice charts, and any geometry homework with a quadrant in it.
Enter the radius in any length unit to get the area, arc length, and perimeter of a quarter circle instantly.
Three short formulas, all built from the radius and the constant π (about 3.14159).
area = ¼ × π × r²The area is one-fourth of a full circle's πr². The arc length is the curved edge — a quarter of the full circumference 2πr, which simplifies to ½πr. The perimeter is the whole boundary: that curved arc plus the two straight radii that meet at the corner, so perimeter = ½πr + 2r. Note that the perimeter is not just the arc — a quarter circle has two straight sides as well.
Suppose you have a quarter circle with a radius of 5.
Arc length
½ × π × 5 = 7.853982 — the curved quarter of the circumference.
Perimeter
7.853982 + 2 × 5 = 17.853982 — the arc plus the two straight radii.
Area
¼ × π × 5² = 19.634954 square units — a quarter of the full disc.
The three outputs answer three different everyday questions. The area (about 19.634954 square units for r = 5) is the flat surface the quadrant covers — the turf in a fan-shaped garden bed, the wood in a rounded shelf corner, the paint for a quarter-round patio. The single most useful insight is that a quarter circle is exactly one-fourth of the matching full circle: four of them snap together into a complete disc, so the area is always πr² ÷ 4 and the arc is always the circumference ÷ 4. The arc length (about 7.853982 here) is just the curved edge, the length of edging or trim you would run along the round side. The perimeter (about 17.853982) is the entire boundary — and it is bigger than the arc alone because a quarter circle has two straight radii meeting at the corner; forgetting those two sides is the most common mistake, so use the perimeter, not the arc, when you fence or frame the whole shape. π is the thread tying it together: the same constant links the radius to both the curved length and the area of every quarter circle, large or small.
The formulas are exact, but a couple of practical points are worth keeping in mind.
True quarter circles and consistent units
These formulas describe a true quarter circle — a clean 90° sector of a perfect circle, with two equal straight radii and one circular arc. A sector with a different angle, an ellipse quadrant, or a rounded corner that is not exactly a quarter of a circle will differ from the computed value. The radius is also unit-agnostic, so the answers are only meaningful if you keep one unit throughout: a radius in centimetres gives an area in square centimetres and an arc length in centimetres, never a mix.