Book A Free Math Class # 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](/content/math/geometry/area-of-a-circle/index.html), the semicircle is one division away. | Symbol | Meaning | Units | | --- | --- | --- | | r | Radius (center to arc) | length (cm, m) | | d | Diameter, the flat edge (d=2r) | length (cm, m) | | A | Area of the semicircle | square units (cm²) | | P | Perimeter of the semicircle | length (cm, m) | | π | About 3.14159 | none | ## 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](/content/math/geometry/sector-of-a-circle/index.html) and to the broader family of circle theorems in the [circles](/content/math/geometry/circles/index.html) 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.