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Semicircle — Definition, Area, and Perimeter Formula

What Is A Semicircle?

A semicircle is half of a circle, formed when a circle is divided into two equal parts by a diameter. The straight side of a semicircle is that diameter; the curved side is exactly half the circle's boundary, called the arc.

Every semicircle is a half-disc with one curved edge and one straight edge. Because the cut is made along the diameter, the two halves are identical, and each is a closed figure bounded by a straight line and a curve. The radius r — the distance from the center to the arc — is the single measurement that determines everything else about it.

What Is The Area of A Semicircle?

The area of a semicircle is half the area of the full circle it came from. Since a circle's area is πr², halving it gives:

A=πr²/2

This is the one place where "half a circle means half" works cleanly. Area measures the flat region inside, and slicing the disc along the diameter splits that region into two equal pieces. If you know the area of a circle, the semicircle is one division away.

What Is The Perimeter of A Semicircle?

Here the straight edge matters. The perimeter of a semicircle is the curved arc plus the flat diameter — not just half the circle's circumference.

The curved part is half the circumference, 1/2(2πr)=πr. The flat part is the diameter, 2r. Add them:

P=πr + 2r = r(π + 2)

A common wrong move is to write P=πr alone, halving the circle's circumference and forgetting the cut. The flat edge is real boundary — your finger traces it when you go around the shape — so it must be counted. This is the single most-missed point on semicircle problems.

The Angle In A Semicircle Is Always A Right Angle

One property makes semicircles especially useful: any angle drawn from the two ends of the diameter to a point on the arc is exactly 90°. This is Thales' theorem, named for the early Greek thinker who is credited with it.

Pick the two endpoints of the diameter, call them A and B, and any point C on the arc. Then ∠ACB=90°, no matter where C sits on the arc. The diameter acts as the hypotenuse of a right triangle for every such point. This is why a semicircle is a quick tool for constructing right angles.

Examples of Semicircle

These build from a single area substitution to a multi-step real-world problem. Each step sits on its own line.

Example 1

Find the area of a semicircle with radius 6 cm. Use π≈3.14.

A=πr²/2 A=3.14×6²/2 A=3.14×36/2 A=113.04/2 = 56.52 cm²

Final answer: 56.52 cm².

Example 2

Find the perimeter of a semicircle with radius 7 cm.

P=πr + 2r P=22/7×7 + 2×7 P=22 + 14 = 36 cm

Final answer: 36 cm.

Example 3

The diameter of a semicircle is 14 cm. Find its area.

r=d/2=14/2=7 cm

A=πr²/2 A=22/7×7²/2 A=22/7×49/2=154/2=77 cm²

Final answer: 77 cm².

Example 4

The area of a semicircle is 100π cm². Find its radius.

πr²/2 = 100π r² = 200 r=√200 = 10√2 ≈ 14.14 cm

Final answer: approximately 14.14 cm.

Example 5

A semicircular window has a radius of 0.5 m. Find the length of weather-stripping needed to seal its full edge.

P=πr + 2r P=3.14×0.5 + 2×0.5 P=1.57 + 1 = 2.57 m

Final answer: 2.57 m.

Example 6

A running track has two straight sides of 80 m joined by two semicircular ends, each of radius 35 m. Find the area enclosed by the two semicircular ends.

A=πr² A=22/7×35² A=22/7×1225 = 3850 m²

Final answer: 3850 m².

Where The Semicircle Earns Its Place

A semicircle is not only half a shape — it is a building block. Two semicircles capping a rectangle make the classic athletics track and the stadium outline. Arched windows and doorways are semicircles set on a straight lintel, and Roman builders relied on the semicircular arch precisely because its geometry distributes load evenly to the supports.

The right-angle property pulls real weight too. Because any point on the arc sees the diameter at 90°, a semicircle gives a reliable way to construct or verify a right angle with only a compass and straightedge — a fact that connects this shape to the sector of a circle and to the broader family of circle theorems in the circles hub.

Common Mistakes With Semicircles

Mistake 1: Halving the circumference and calling it the perimeter

Where it slips in: Any perimeter question, because "half a circle" suggests "half the boundary".
Don't do this: Report P=πr and ignore the straight edge.
The correct way: Add the diameter: P=πr + 2r. The student who pictures the shape and traces its outline never drops the flat edge; the one who only manipulates the formula often does. Forgetting the diameter is the first-instinct error on every semicircle perimeter problem.

Mistake 2: Using diameter where the formula wants radius

Where it slips in: When the problem states the diameter but the area and perimeter formulas are written in r.
Don't do this: Substitute the diameter directly into the area formula.
The correct way: Halve the diameter first: r=d/2.

Mistake 3: Mixing up area units and length units

Where it slips in: Reporting a perimeter in square units or an area in plain units.
Don't do this: Write the area as cm or the perimeter as cm².
The correct way: Area is always in square units (cm², m²); perimeter is always in length units (cm, m).

Conclusion

• A semicircle is half a circle, cut along the diameter.
• Its area is πr²/2 — half the circle's area.
• Its perimeter is πr + 2r, the arc plus the diameter, not half the circumference.
• By Thales' theorem, the angle in a semicircle is always 90°.
• One line of symmetry runs perpendicular to the diameter.