How this instrument works
A quarter circle is a 90° wedge cut from a full disk — one of four identical pieces made by slicing a circle along two perpendicular radii. Its area comes straight from that tiling: four such wedges reassemble into the whole disk with nothing left over, so A = πr² ⁄ 4 is exactly one quarter of the familiar πr². Its perimeter is a different animal, part curve and part straight edge, and that mismatch is exactly why a landscaper or fabricator reaching for this shape rarely wants only one figure — order edging and you need the perimeter, order paving or sod and you need the area, and most jobs need both from the same radius in one pass.
The perimeter formula reads as two pieces stitched into one line: 2r for the pair of straight radius edges meeting at the wedge's point, plus πr ⁄ 2 for the curved arc — exactly one quarter of a full circle's circumference 2πr. The straight edges are the part people forget. Mentally merging 'quarter circle' with 'circle' leads to reporting the arc alone as the perimeter, but the two straight sides are real boundary, and skipping them under-measures a fence, a trim strip, or a border by a fixed 2r every time — proportionally worse the smaller the radius.
The two outputs also scale differently as r grows, which is worth noticing on its own: perimeter is linear in r, so doubling the radius exactly doubles the boundary length, while area carries an r² and quadruples on the same doubling. A wedge of radius 8 has exactly twice the perimeter of one with radius 4 but four times the area — the same square-law-versus-linear-law split that shows up in paint coverage and sail drag alike. At the limit r = 0 both figures collapse to zero, the wedge shrunk to a single point.
- Enter your quarter circle's radius into the Radius field — any length unit works, as long as you read the results in that same unit.
- Area updates immediately, showing πr² ⁄ 4 worked to full precision.
- Perimeter updates alongside it, showing 2r + πr ⁄ 2 — the two straight edges plus the curved arc.
- Read both figures straight from the one radius instead of switching between separate area-only and perimeter-only sheets.
Worked example — a quarter circle with radius 4
Take a garden bed shaped like a quarter circle with radius 4 metres, the kind of wedge that fills a square corner. Area comes out to A = π × 4² ⁄ 4 = 16π ⁄ 4 = 4π = 12.566370614359172 square metres, the exact figure this sheet reports at full precision — roughly 12.566 m² for ordering mulch or turf.
Perimeter in the same pass: P = 2×4 + π×4 ⁄ 2 = 8 + 2π = 14.283185307179586 metres, made up of the two 4-metre straight edges plus a 2π ≈ 6.283-metre arc, useful the moment edging or a low wall is needed along the curve. Both answers come from the single radius entered once, matching the exact values a quarter circle of that size must produce.
Questions
What is the formula for the area of a quarter circle?
A = πr² ⁄ 4 — exactly one quarter of a full circle's πr², since four such wedges tile a complete disk with no gaps or overlaps. For radius 4 that gives 4π ≈ 12.566 square units, the same figure this sheet returns at full precision.
Why does the perimeter formula include 2r as well as the arc length?
Because the boundary of a quarter circle is not only the curved edge — it also includes the two straight radius segments that meet at the centre to form the right angle. Skipping the 2r term is the single most common mistake with this shape; it under-counts the perimeter by exactly twice the radius.
How is a quarter circle's perimeter different from a full circle's circumference?
A full circle's circumference is 2πr, purely curved. A quarter circle's perimeter, P = 2r + πr ⁄ 2, mixes a quarter of that curve (πr ⁄ 2) with two straight radius edges the full circle does not have. The curved portion alone is exactly one quarter of 2πr, but the total perimeter is not one quarter of the circumference — the straight edges push it higher.
Does doubling the radius double both the area and the perimeter?
No — only the perimeter doubles, because P is linear in r. Area carries r² and quadruples instead: a radius-8 wedge has exactly twice the perimeter of a radius-4 wedge (28.566 vs 14.283) but four times the area (50.265 vs 12.566). The two quantities follow different scaling laws even though they share one input.
What happens when the radius is zero?
Both area and perimeter collapse to zero — the degenerate case of a wedge with no size at all, where the two straight edges and the arc all shrink to a single point. This sheet reports exactly 0 for both figures rather than an error, since the formulas stay well defined at r = 0.
How is this different from the separate quarter-circle-area and quarter-circle-perimeter calculators?
Those two sheets are single-purpose: one returns only πr² ⁄ 4, the other only 2r + πr ⁄ 2. This one runs both formulas on the same radius in a single pass, which is faster when a job — fencing a curved garden bed, say — needs the edging material and the interior coverage at the same time.