Mathwords: Semicircle

Key Formula

A = \frac{1}{2}\pi r^2 \qquad P = \pi r + 2r

Where:

$A$ = Area of the semicircle
$P$ = Perimeter of the semicircle (curved arc plus diameter)
$r$ = Radius of the full circle

Worked Example

Problem: A semicircle has a radius of 10 cm. Find its area and perimeter.

Step 1: Find the area using half the area of a full circle.

A = \frac{1}{2}\pi r^2 = \frac{1}{2}\pi(10)^2 = 50\pi \approx 157.08 \text{ cm}^2

Step 2: Find the perimeter. The curved part is half the circumference, and the straight part is the diameter.

P = \pi r + 2r = \pi(10) + 2(10) = 10\pi + 20 \approx 51.42 \text{ cm}

Answer: The area is

50\pi \approx 157.08

cm² and the perimeter is

10\pi + 20 \approx 51.42

cm.

Why It Matters

Semicircles appear frequently in architecture (arched doorways and windows) and engineering (cross-sections of tunnels). Thales' theorem states that any angle inscribed in a semicircle is a right angle, which is a foundational result in geometry used to construct perpendicular lines.

Common Mistakes

Mistake: Forgetting to include the diameter when calculating the perimeter of a semicircle.

Correction: The perimeter consists of both the curved arc (

\pi r

) and the straight diameter (

2r

). Simply halving the circumference gives you only the arc length, not the full perimeter.

Related Terms

Circle — A semicircle is half of a circle
Arc of a Circle — A semicircle is a specific 180° arc
Diameter — The straight edge that bounds a semicircle
Circumference — Half the circumference gives the arc length