Semicircle Area Calculator

A semicircle is exactly half of a disk, so its area is half the full circle: A = (1/2) x pi x r^2. For r = 4, A = (1/2) x pi x 16 = 8 pi, about 25.13 square units. The same number falls out of the general sector formula A = (1/2) r^2 theta when theta = pi radians (180 degrees), because plugging in theta = pi reduces it to (1/2) pi r^2. The calculator also reports the full-disk area (pi r^2) next to the semicircle area as a quick cross-check: the full circle should always be exactly twice the semicircle.

Perimeter means the total length of the closed boundary of a shape. A full circle's boundary is the circumference and nothing else, so 2 pi r is both arc length and perimeter. A semicircle is bounded by two pieces: the curved arc (length pi r) and the straight diameter that caps it off (length 2r). Drop the diameter and you no longer have a closed region, just an open arc. Adding both gives P = pi r + 2r = r (pi + 2), which is roughly 5.1416 r. If you only need the curved side (for fencing the rounded edge of a flowerbed, say, where the straight side is a wall), use arc length pi r on its own, not perimeter.

The centroid (geometric center of mass for a uniform-density flat semicircular plate) sits on the axis of symmetry, perpendicular to the diameter, at distance 4r / (3 pi) above the midpoint of the diameter. That works out to roughly 0.4244 r. For r = 1, the centroid is about 0.4244; for r = 3, about 1.273. The result comes from Pappus's centroid theorem (and from a direct integral): swing the semicircular area around its diameter and you sweep out a sphere of volume (4/3) pi r^3; the theorem ties that volume to the centroid distance via V = 2 pi d x A, which solves to d = 4r / (3 pi). The same constant appears in moment-of-area calculations for half-round beams and in CNC toolpath planning for half-disk pockets.

Semicircle geometry appears in half-round ducts, gutters, arches, and tunnel cross sections. Area helps estimate cross-sectional capacity, while arc length and perimeter help estimate lining or sheet material. For structural calculations, pi r^4 / 8 is the area moment of inertia about the diameter axis. A centroidal-axis value is different and requires the parallel-axis theorem.

Arc length for any circular arc is s = r theta when the angle is in radians. A semicircle subtends half a full turn, which is theta = pi radians (the radian measure of 180 degrees), so s = r x pi = pi r exactly. No approximation. The full-circle circumference 2 pi r is two semicircle arcs back to back, confirming the half-turn split. If you write the same answer in degrees you get s = (theta / 180) x pi x r = (180 / 180) x pi x r = pi r, the same value: the formula is unit-aware but the geometry isn't. This is also why pi was historically defined as the ratio of circumference to diameter (or equivalently, semicircle arc length to radius), so pi r is the most fundamental length the constant produces.