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: a 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.

A square inscribed in a circle of radius r has side r√2 and area 2r², so half of it is r². Set that against the semicircle’s 1.571r² and the curved half wins by 57%. The same 57% shows up between the whole circle and the whole inscribed square, which is a useful number to carry: rounding a square opening loses you more than a third of its area, and squaring a round one gains you a third back.

How to tell a true semicircle from a segmental arch

Measure the opening across, then measure the rise from the springing line to the top of the curve. If the rise is exactly half the width, the arch is a semicircle. Anything less is a segmental arch on a larger radius, and anything more is a horseshoe.

That single check matters because the two need different formulas. A semicircle is ½πr². A segmental arch is a circular segment, and its area comes from ½r²(θ − sin θ) with a radius you have to derive from the chord and rise first.

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. ✓ That is why a stadium shape, a rectangle with a semicircular cap on each end, has exactly one circle’s worth of area in its two ends.

The perimeter is a separate calculation, πr + 2r, and it does NOT split as cleanly as the area does. A semicircle carries 82% of the parent circle’s perimeter but exactly 50% of its area. That page covers why.