A semicircle is exactly half a circle, so its curved arc is half the full circumference. The area formula is πr²/2 and the full perimeter — the curved arc plus the flat diameter — is πr + 2r. These two formulae are all you need for any semicircle question at GCSE.
What is a semicircle?
A semicircle is the region bounded by a diameter and the arc that it cuts off. It has:
- A curved edge (the arc) of length equal to half the circumference of the full circle.
- A straight edge (the diameter) of length 2r.
- An area equal to half the area of the full circle.
The radius r is the distance from the centre of the diameter to any point on the curved arc.
What is the area formula for a semicircle?
The full circle has area πr². A semicircle is half of this:
Area of semicircle = πr² / 2
Worked example 1: Find the area of a semicircle with radius 5 cm. Give your answer to 1 decimal place.
Area = π × 5² / 2 = π × 25 / 2 = 12.5π ≈ 39.3 cm²
Worked example 2: A semicircle has diameter 14 cm. Find its area. Leave the answer in terms of π.
Diameter = 14 cm, so radius = 7 cm.
Area = π × 7² / 2 = 49π/2 cm² = 24.5π cm²
What is the perimeter formula for a semicircle?
The perimeter of a semicircle includes both the curved arc and the straight diameter.
- Curved arc = half the full circumference = (2πr) / 2 = πr
- Straight diameter = 2r
Perimeter of a semicircle = πr + 2r
This can be factorised as r(π + 2) if the question asks for an exact form.
Worked example 3: Find the perimeter of a semicircle with radius 6 cm. Give your answer to 1 decimal place.
Perimeter = π × 6 + 2 × 6 = 6π + 12 ≈ 18.85 + 12 = 30.8 cm (1 d.p.)
Worked example 4: A semicircle has diameter 20 cm. Find the perimeter. Leave in terms of π.
Radius = 10 cm. Perimeter = 10π + 20 cm.
How do you find the area and perimeter of a composite shape involving a semicircle?
Worked example 5: A shape consists of a rectangle (length 10 cm, width 6 cm) with a semicircle placed on one of the longer sides. Find the area and perimeter of the whole shape.
The semicircle's diameter = 10 cm, so radius = 5 cm.
Area:
- Rectangle: 10 × 6 = 60 cm²
- Semicircle: π × 5² / 2 = 12.5π cm²
- Total area = 60 + 12.5π ≈ 60 + 39.27 ≈ 99.3 cm² (1 d.p.)
Perimeter: The perimeter goes round the outside: two short sides of the rectangle (6 + 6), one long side of the rectangle (10), and the curved arc of the semicircle. The long side where the rectangle meets the semicircle is internal and is NOT part of the perimeter.
- Two short sides: 2 × 6 = 12 cm
- One long side: 10 cm
- Curved arc: π × 5 = 5π cm
- Total perimeter = 22 + 5π ≈ 22 + 15.71 ≈ 37.7 cm (1 d.p.)
| Part | Measurement |
|---|---|
| Two widths of rectangle | 12 cm |
| One length of rectangle | 10 cm |
| Semicircular arc (r = 5) | 5π cm ≈ 15.7 cm |
| Total | ≈ 37.7 cm |
What formula gives an exact answer?
If the question says "leave your answer in terms of π", do not use a decimal approximation for π. Simply carry the symbol through your working:
Area of semicircle with r = 5: 25π/2 cm² (exact)
Perimeter with r = 5: (5π + 10) cm or 5(π + 2) cm (exact)
What mistakes should you avoid?
Mistake 1 — Forgetting to include the diameter in the perimeter. The perimeter of a complete semicircular shape includes the straight diameter. Students who find only the arc length (πr) are finding just the curved part — always add 2r for the diameter unless the shape is attached to something else along that straight edge.
Mistake 2 — Using the diameter instead of the radius in the area formula. The formula πr²/2 needs the radius r. If given the diameter d, divide by 2 first: r = d/2. Substituting the diameter directly into the formula quadruples the area.
Mistake 3 — Including an internal straight edge in the perimeter of a composite shape. When a semicircle is joined to a rectangle, the straight edge where they meet is interior and does not contribute to the outer perimeter.
Frequently asked questions
How is the area of a semicircle different from a half-sector?
A semicircle is bounded by a diameter (a chord through the centre). A half-sector is bounded by two radii and an arc — it is a "pie slice" of half the circle (angle 180°). For a 180° sector, the two radii form a straight line (the diameter), so a half-sector and a semicircle are actually the same shape. Other sectors (not 180°) are smaller "pie slices".
What is the angle subtended by a semicircle at the circumference?
By the inscribed angle theorem (a circle theorem tested at GCSE Higher), the angle in a semicircle is always 90°. Any angle formed at a point on the curved arc of a semicircle, with its two arms going to the ends of the diameter, equals 90°. This is Thales' theorem.
How do you find the radius if you are given the area of a semicircle?
Rearrange the formula: A = πr²/2 → r² = 2A/π → r = √(2A/π). For example, if A = 50 cm²: r² = 100/π ≈ 31.83, so r ≈ √31.83 ≈ 5.64 cm.
Can a semicircle question appear in the non-calculator paper?
Yes — you may be asked to leave the answer in terms of π, which avoids needing a decimal approximation. For example, "Find the exact area of a semicircle of radius 4 cm" expects the answer 8π cm², written without a decimal.
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