Semicircle Area Calculator

A semicircle is exactly half a circle — cut along a diameter. The area is half the area of the full circle:

A = ½ × π × r²

Equivalent forms:

• From diameter d = 2r: A = π × d² / 8
• From circumference of the full parent circle C: A = C² / (8π)

Worked example — half-moon window: A half-round transom window above a doorway has a 48-inch wide opening (the diameter). r = 24 in. A = 0.5 × π × 576 ≈ 904.78 sq in ≈ 6.28 sq ft of glass.

If you’re ordering glass, expect to pay by the cut shape (more expensive than a square piece of the same area, because curved cuts waste material).

Where semicircles show up:

• Half-moon transom windows. The decorative glass panel above doors and tall windows.
• Arched doorways and gateways. The curved top section of a Norman or Roman arch is a semicircle.
• Dome floor plans. A hemispherical dome viewed from above is a circle, but the cross-section is a semicircle.
• Half-pipe tunnels and culverts. Drainage half-pipes and tunnel cross-sections.
• Decorative garden beds. Curved beds along a fence or wall.
• Theater stage aprons. Half-round stage extensions into the audience.

Comparing semicircles to other “half” shapes:

• Half a square (cut diagonally) = a right isosceles triangle with area ½ × s².
• Half a square (cut across the middle) = a rectangle with area ½ × s² (different aspect ratio than the triangle).
• Half a circle = ½ × π × r² ≈ 1.5708 × r².

For a square inscribed in a circle of radius r, the half-square triangle has area r² (slightly less than the semicircle’s ~1.57r²). The circle holds about 57% more area than the inscribed square. Geometry trivia.

The “arch test”:

If you can stand inside an opening at its center and your fingertips just brush the curve at the top, then the opening is a semicircle. (Approximately — assumes your arm span equals your height. Vitruvian assumption.)

Sanity check:

• r = 0: A = 0. ✓
• A semicircle has half the area of a full circle: ½ × πr² ✓.
• Two semicircles back to back make one full circle. ✓ (See the stadium shape — the two end caps combine to one full circle.)

The perimeter of a semicircle is a separate calculation (πr + 2r) — covered on the semicircle perimeter page.