Calculate the area, perimeter, arc length, radius, diameter, and centroid of a semicircle. Solve from radius, diameter, area, or perimeter with unit conversion and presets.
A semicircle is exactly half of a circle, created by cutting a circle along a diameter. Despite its simplicity, the semicircle appears everywhere — in architecture (arched windows, doorways, bridges), engineering (half-pipe channels, cross-sections of tunnels), and mathematics (Thales' theorem states that any angle inscribed in a semicircle is a right angle).
The area of a semicircle with radius r is πr²/2, exactly half the area of the full circle. The perimeter (total boundary length) is more nuanced: it consists of the curved arc (πr) plus the straight diameter (2r), giving a total perimeter of πr + 2r = r(π + 2). Note that the "perimeter" and the "arc length" are different — the arc is only the curved part.
The centroid (center of mass) of a semicircle lies at a height of 4r/(3π) above the diameter, approximately 0.4244r. This is important in structural engineering for calculating moments and bending stresses in semicircular cross-sections.
This calculator lets you solve in any direction: enter the radius, diameter, area, or perimeter, and it computes all other properties. It includes full-circle comparisons, visual proportion bars, and a reference table so you can quickly look up values for common semicircle sizes.
Semicircle problems often start from different known values. One drawing gives a diameter, another specification lists an area, and a fabrication note may give the full perimeter including the flat base. This calculator is useful because it works in all of those directions and converts them back to the common geometric properties without forcing you to rearrange formulas each time.
It is also practical for real shapes, not just textbook diagrams. Semicircular windows, tunnels, arches, gutters, and half-pipe sections all depend on the same measurements, but the quantity you need may change from job to job. Seeing the radius, diameter, arc length, area, perimeter, and centroid in one place makes it easier to move from geometry to layout, material estimates, or structural interpretation.
Area: A = πr²/2 Perimeter: P = πr + 2r = r(π + 2) Arc Length: L = πr Diameter: d = 2r Centroid height: ȳ = 4r/(3π) Reverse: Radius from area: r = √(2A/π) Radius from perimeter: r = P/(π + 2)
Result: Radius = 15 cm, area ≈ 353.43 cm², arc length ≈ 47.12 cm, perimeter ≈ 77.12 cm
If the diameter is 30 cm, the radius is 15 cm. The semicircle area is $pi r^2 / 2 = pi imes 225 / 2 approx 353.43$ cm². The curved arc is $pi r approx 47.12$ cm, and the full perimeter adds the straight diameter: $47.12 + 30 approx 77.12$ cm. The centroid sits $4r/(3pi) approx 6.37$ cm above the diameter, which is useful when treating the semicircle as a physical cross-section rather than just a sketch.
A semicircle is simple to picture, but one detail causes repeated mistakes: its boundary is made of a curved arc and a straight diameter. That means the perimeter is not just half of a circumference. You must add the flat edge. This calculator keeps the arc length and full perimeter separate so you can tell immediately whether you are measuring only the curve or the entire outline.
Many real problems are reverse problems. You may know the area of a semicircular pond liner, the span of an arch, or the perimeter of a half-round part and need to recover the radius. Because the formulas can be rearranged cleanly, the calculator lets you solve from radius, diameter, area, or perimeter and then expands that value into the full geometry. That is especially useful when checking shop drawings or back-solving a missing dimension from a known specification.
The centroid of a semicircle is not at the circle center and not halfway up the radius. It sits at $4r/(3pi)$ above the diameter. That makes a difference in structural and mechanical work because the centroid location affects bending, balance, and section properties. By reporting it alongside area and perimeter, the calculator becomes more useful for engineering intuition, not just for basic mensuration.
The area of a semicircle with radius r is A = πr²/2. It is exactly half the area of a full circle with the same radius.
The perimeter is P = πr + 2r. The πr part is the curved arc, and 2r is the straight diameter edge. Total: r(π + 2) ≈ 5.14r.
Rearrange A = πr²/2 to get r = √(2A/π). For example, if A = 50, then r = √(100/π) ≈ √31.83 ≈ 5.64.
The arc length (curved part only) is L = πr. This is half the circumference of the full circle (2πr).
The centroid is at a height of 4r/(3π) ≈ 0.4244r above the flat diameter edge. It is NOT at the center of the radius.
Yes, a semicircle is defined as exactly half of a circle, divided by the diameter. Its area, arc length, and central angle (180°) are all exactly half.