Area of a Semi-Circle Calculator
The area of a semi-circle is half the area of the full circle it was cut from: A = (1/2) x pi x r^2. This calculator starts from whichever measurement you actually have, the radius, the diameter, the area itself, the perimeter, or just the curved arc length, solves for the radius, and then fills in every other measure with the steps shown.
Most problems do not hand you the radius. They hand you the flat side: the diameter, because that is the edge that sits on a windowsill, spans an arch, or borders a garden path. So the worked example on this page starts from a diameter of 14, and the first step is the one real work demands: halve it before anything gets squared.
Units stay consistent: enter the radius in feet and the area is in square feet. The diameter is a real edge of the shape, so the perimeter includes it.
Quick Answer
Key takeaways
- Halve the diameter before you square anything. Squaring the diameter and then halving gives four times too much area.
- Perimeter is not half the circumference. The full boundary is pi x r + 2r: the curved arc plus the straight diameter.
- Any single measure determines all the others. This calculator solves backwards from area, perimeter, or arc length to the radius, then forwards again.
- A semicircle is exactly half a disc, never an oval or a half-ellipse. Different shape, different formula.
- Quick Answer
- The semicircle formulas, in one place
- How to use this semicircle calculator
- Worked example: diameter 14, step by step
- The perimeter trap, and the diameter trap
- Reference table: area and perimeter by radius
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
The semicircle formulas, in one place
A semicircle is a disc cut along a diameter. Every formula for it is the full-circle formula with the right adjustment: half for the area and the curved edge, plus the diameter itself for the perimeter.
How to use this semicircle calculator
Tell it what you know, not what the formula wants. The mode menu is the whole interface.
Pick the measurement you have. Radius and diameter are the common cases in homework. Perimeter shows up in fencing and edging problems, arc length in machining and track problems, and area when you are working backwards from a material quantity.
Enter the value and calculate. The steps box shows the radius being recovered first, then each measure being built from it, so a backwards solve is as checkable as a forwards one.
Pick the output your problem actually asks for. Area, arc, and perimeter are three different numbers (76.969, 21.9911, and 35.9911 in the example) and exam questions swap them deliberately.
Worked example: diameter 14, step by step
The calculator is pre-filled with a diameter of 14. Step one: r = 14 / 2 = 7. Step two: A = (1/2) x pi x 7^2 = (1/2) x pi x 49 = 76.969. Step three: arc = pi x 7 = 21.9911, and perimeter = 21.9911 + 14 = 35.9911.
| Measure | Value (diameter 14) |
|---|---|
| Radius | 7 |
| Area | 76.969 square units |
| Arc length (curved edge) | 21.9911 |
| Perimeter (arc + diameter) | 35.9911 |
| Full circle area | 153.938 square units |
The curved edge deserves its own note. The arc of a semicircle is pi x r, which is the 180 degree case of the general arc formula s = r x theta. The arc length calculator handles every other angle, and it is worth seeing the two side by side: at 180 degrees its general formula collapses to exactly the pi r used here.
The perimeter trap, and the diameter trap
Two mistakes account for most wrong semicircle answers. Both are one-line errors, and both survive because the wrong answer looks plausible.
The perimeter trap
- Wrong: perimeter = half the circumference = pi r = 21.9911.
- Right: the boundary includes the flat diameter edge, so P = pi r + 2r = 35.9911.
- The short version understates fencing or trim needs by 14 units here, about 39 percent.
- Exception that proves the rule: a semicircle built against a wall needs only the arc fenced, because the wall supplies the diameter.
The diameter trap
- Wrong: A = (1/2) pi x 14^2 = 307.876, using the diameter as the radius.
- Right: halve first, r = 7, then A = (1/2) pi x 49 = 76.969.
- The error multiplies the answer by exactly 4, because the diameter is squared before the halving happens.
- Written halving as its own step is the cheapest error check in this whole topic.
Reference table: area and perimeter by radius
Area grows with the square of the radius while perimeter grows in direct proportion, which is why doubling the size quadruples material but only doubles edging.
| Radius | Diameter | Area | Arc | Perimeter |
|---|---|---|---|---|
| 1 | 2 | 1.5708 | 3.1416 | 5.1416 |
| 2 | 4 | 6.2832 | 6.2832 | 10.2832 |
| 5 | 10 | 39.2699 | 15.708 | 25.708 |
| 7 | 14 | 76.969 | 21.9911 | 35.9911 |
| 10 | 20 | 157.0796 | 31.4159 | 51.4159 |
Semicircles turn up in semicircular garden beds and patios, arched window glass, half-round gutters, and the ends of running tracks and capsule shapes. In every one of those jobs the quantity ordered (glass, pavers, turf) is the area, while the trim, edging, or gutter run is the perimeter, and the two numbers answer to different formulas. Do not confuse the shape with an ellipse cut in half either: a true semicircle has one radius, while an oval has two different semi-axes, which is why the area of an oval calculator uses pi x a x b instead.
What competing calculators miss
Winning move 1: start from any measure. The common tools (Brave Calculator's semicircle pages are typical) accept the radius only. Real problems arrive as a diameter, a perimeter, or a target area, and each of those needs a backwards solve this page does in the same box.
Winning move 2: the traps named and priced. Dash Calculator explains the perimeter trap well in prose, but its tool still takes radius only. Here the perimeter trap and the diameter trap are laid out with the exact wrong answers they produce on the worked example, so you can recognize them in your own working.
Winning move 3: steps for backwards solves. Recovering r = P / (pi + 2) from a perimeter is the step students cannot find in most guides. It is printed, with your numbers, in the steps box.
Frequently asked questions
The arc is only the curved edge: pi x r, which is 21.9911 at radius 7. The perimeter is the whole boundary, arc plus the straight diameter: pi r + 2r = 35.9911. Fencing a free-standing semicircular bed needs the perimeter; the curved side alone is the arc.
Halve the diameter to get the radius, then use A = (1/2) pi r^2. Diameter 14 gives r = 7 and A = (1/2) x pi x 49 = 76.969. The costly mistake is squaring the diameter first: that answer is four times too large. Pick Diameter in this calculator and the halving is done as the first printed step.
Yes: r = sqrt(2A / pi). An area of 76.969 gives r = sqrt(153.938 / pi) = sqrt(49) = 7. Pick Area in the mode menu and the calculator runs this backwards solve, then rebuilds the diameter, arc, and perimeter from the recovered radius.
Any consistent unit. Enter lengths in feet and the area is square feet; enter centimetres and the area is square centimetres. The calculator does not convert between systems, so measure the diameter and read the perimeter in the same unit you measured in.
Yes, the words name the same shape: exactly half of a disc, cut along a diameter. It is not the same as half of an oval or ellipse. A semicircle has a single radius in every direction from the center of its flat edge; an ellipse has a long and a short semi-axis, and its half follows a different area formula.
As a check. The semicircle area must be exactly half the full circle area for the same radius, so if the two numbers in the result do not stand in a 2-to-1 ratio, an input was mistyped. At radius 7 the pair is 76.969 and 153.938.
Split it and add. A common layout is a rectangle of width equal to the diameter with a semicircle on one end: total area = (length x width) + (1/2) pi r^2, with r = width / 2. Compute the semicircle part here, the rectangle part by hand, and add them. Running tracks and capsule floor plans are built exactly this way.
The tool uses full double-precision pi (3.141592653589793), not the classroom 3.14. With pi = 3.14 the worked example gives 76.93 instead of 76.969, a small gap that matters when the area drives a material order. Results are displayed to 4 decimal places.
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References
- Wolfram Research, "Semicircle, Wolfram MathWorld". MathWorld: Semicircle. Accessed October 2026.
- Wolfram Research, "Circular Segment, Wolfram MathWorld". MathWorld: Circular Segment. Accessed October 2026.
- Wolfram Research, "Circular Sector, Wolfram MathWorld". MathWorld: Circular Sector. Accessed October 2026.
- Wolfram Research, "Arc Length, Wolfram MathWorld". MathWorld: Arc Length. Accessed October 2026.
- OpenStax, Rice University, "Precalculus 2e: areas and perimeters of circular figures". OpenStax: Precalculus 2e. Accessed October 2026.