Semicircle Formula: Area, Perimeter and Arc Length Explained

The semicircle formula for area is A = (π × r²) ÷ 2, and for perimeter it is P = πr + 2r, where r is the radius. A semicircle is exactly half a circle, so the formulas come straight from the circle ones, with one adjustment for the flat edge that most people forget.

The semicircle formulas at a glance

What you want	Formula	Needs
Area	A = ½ × π × r²	Radius
Area from the diameter	A = π × d² ÷ 8	Diameter
Curved edge (arc length)	πr	Radius
Straight edge (diameter)	d = 2r	Radius
Perimeter	P = πr + 2r = r(π + 2)	Radius
Radius from the area	r = √(2A ÷ π)	Area

Why the area is half of a circle

A full circle has area πr². Slice it along any diameter and the two pieces match exactly, so each piece holds half the area. That is all the area formula says. If you can remember the circle formula, you can always rebuild the semicircle one without learning it separately.

Using the diameter instead

Sometimes a drawing gives the diameter rather than the radius. Since r = d ÷ 2, we get r² = d² ÷ 4, so the area becomes ½ × π × d² ÷ 4, which is π × d² ÷ 8. The two forms always agree; pick whichever matches the number you are given, and you avoid an extra halving step.

Why the perimeter is not simply half the circumference

This is the classic slip. The circumference of a full circle is 2πr, so the curved part of the semicircle is half of that: πr. But the shape is closed by a straight line, the diameter, and that length counts as part of the boundary. The perimeter is therefore the curve plus the straight edge: πr + 2r. If you only halve the circumference, you leave out the diameter and your answer will be too short.

The same idea applies to anything built around a semicircle, such as a window arch, a garden bed or a running track end. If you are fencing only the curved side, you want the arc πr. If you are edging the whole shape, you want the full perimeter.

Worked examples

1. Radius 10 cm. Area = ½ × π × 10² = ½ × π × 100 ≈ 157.1 cm². Arc = π × 10 ≈ 31.4 cm. Perimeter = 31.4 + 20 = 51.4 cm.
2. Diameter 12 m. The radius is 6 m. Area = ½ × π × 36 ≈ 56.5 m². Using the diameter form: π × 144 ÷ 8 ≈ 56.5 m², the same answer. Perimeter = π × 6 + 12 ≈ 18.85 + 12 = 30.85 m.
3. Perimeter known: 25.7 cm. Use P = r(π + 2), so r = 25.7 ÷ 5.1416 ≈ 5 cm. Area = ½ × π × 25 ≈ 39.3 cm².
4. Area known: 100 cm². r = √(2 × 100 ÷ π) = √63.66 ≈ 7.98 cm. So the diameter is about 15.96 cm.

A practical example: a semicircular garden bed

Suppose a bed is a semicircle with a straight edge of 4 m against a wall. The diameter is 4 m, so r = 2 m. The area to cover with soil is ½ × π × 4 ≈ 6.28 m². The edging for the curved side is π × 2 ≈ 6.28 m (a coincidence of the numbers, since πr and ½πr² match when r = 2). If you also need to trim the wall side, add 4 m. Note that you are doing a unit conversion if you buy by the litre or bag; for example a depth of 10 cm over 6.28 m² is 0.628 m³, which is about 628 litres.

Common mistakes to avoid

• Forgetting the diameter in the perimeter. Halving 2πr leaves the straight side out.
• Using the diameter as the radius. This quadruples the area. Check which number the question gives.
• Mixing units. If one length is in centimetres and another in metres, convert before calculating.
• Rounding too early. Keep π on the calculator until the last step, or use at least 3.1416.
• Using the wrong power. Area uses r squared; perimeter uses r to the first power. Area units are squared, perimeter units are not.

Where semicircle formulas turn up

You will meet them in school geometry, in architecture (arched windows and doorways), in landscaping, in carpentry and in engineering, where a semicircular cross-section such as a half-pipe is common. They also show up when you break a complex shape into simple parts: a running track, for example, is two straight sections and two semicircular ends. Unit conversion often rides along with the geometry, whether that is millilitres in a half-cylinder or degrees; if you like this sort of practical measuring, our guides to the freezing point in Fahrenheit and the Australian pint in ml use the same step-by-step conversion habits. For the underlying geometry, see Wikipedia’s page on the semicircle.

Quick method for any semicircle problem

1. Write down what you are given: radius, diameter, area or perimeter.
2. Convert to the radius if needed.
3. Decide what the question wants: area, arc only or full perimeter.
4. Apply the matching formula and keep units consistent.
5. Check the answer is sensible, for example that the area is about half of the corresponding circle.

Frequently asked questions

Conclusion

Remember that the semicircle is half a circle for area, but half a circle plus a straight edge for perimeter. Keep the radius clear, keep your units consistent, and you can rebuild every semicircle formula from the circle ones whenever you need them.