Semicircle Area & Perimeter Calculator

About the Semicircle Area & Perimeter Calculator

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.

When This Page Helps

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 together makes it easier to move from geometry to layout, material estimates, or structural interpretation.

How to Use the Inputs

1. Choose what you know: radius, diameter, area, or perimeter.
2. Select a measurement unit.
3. Enter the known value.
4. Or click a preset to load common semicircle sizes.
5. View all computed properties: area, perimeter, arc length, centroid, and more.
6. Compare the semicircle to its parent full circle.
7. Use the reference table to look up values for standard radii.

Example Calculation

Tips & Best Practices

• The perimeter of a semicircle is NOT half the circumference — you must add the diameter (straight edge) to the arc length.
• Thales' theorem: any triangle inscribed in a semicircle with the hypotenuse as the diameter is a right triangle.
• For a half-pipe or trough, the cross-sectional area is a semicircle — use this to calculate flow capacity.
• The ratio of arc length to diameter in a semicircle is π/2 ≈ 1.571, always the same regardless of size.
• A semicircular arch distributes load evenly, which is why it has been used in bridges and aqueducts for millennia.

The Semicircle Has Two Different Boundaries

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.

Solving Backward From Area Or Perimeter

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.

Why The Centroid Matters

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.

Frequently Asked Questions

• What is the area of a semicircle?

  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.

• What is the perimeter of a semicircle?

  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.

• How do I find the radius from the area?

  Rearrange A = πr²/2 to get r = √(2A/π). For example, if A = 50, then r = √(100/π) ≈ √31.83 ≈ 5.64.

• What is the arc length of a semicircle?

  The arc length (curved part only) is L = πr. This is half the circumference of the full circle (2πr).

• Where is the centroid of a semicircle?

  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.

• Is a semicircle always exactly half of a circle?

  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.