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 $\pi r^2$, halving it gives:
$$A = \frac{\pi r^2}{2}$$
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.
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) |
$\pi$ | 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, $\frac{1}{2}(2\pi r) = \pi r$. The flat part is the diameter, $2r$. Add them:
$$P = \pi r + 2r = r(\pi + 2)$$
A common wrong move is to write $P = \pi r$ 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 $\angle ACB = 90°$, 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 $\pi \approx 3.14$.
$$A = \frac{\pi r^2}{2}$$ $$A = \frac{3.14 \times 6^2}{2}$$ $$A = \frac{3.14 \times 36}{2}$$ $$A = \frac{113.04}{2} = 56.52 \text{ cm}^2$$
Final answer: 56.52 cm².
Example 2
Find the perimeter of a semicircle with radius 7 cm. A student answers $\pi r = 22$ cm. What did they miss, and what is the correct perimeter? Use $\pi \approx \dfrac{22}{7}$.
The first instinct is to take half the circle's circumference and stop: $\pi r = \frac{22}{7} \times 7 = 22$ cm. But trace the boundary with a finger — after the curved top you still have to cross back along the flat cut. That straight edge, the diameter, was left out.
The full perimeter adds the diameter:
$$P = \pi r + 2r$$ $$P = \frac{22}{7} \times 7 + 2 \times 7$$ $$P = 22 + 14$$ $$P = 36 \text{ cm}$$
Final answer: 36 cm. The flat edge is what the half-circumference shortcut forgets.
Example 3
The diameter of a semicircle is 14 cm. Find its area. Use $\pi \approx \dfrac{22}{7}$.
First convert the diameter to a radius:
$$r = \frac{d}{2} = \frac{14}{2} = 7 \text{ cm}$$
Then apply the area formula:
$$A = \frac{\pi r^2}{2}$$ $$A = \frac{\frac{22}{7} \times 7^2}{2}$$ $$A = \frac{\frac{22}{7} \times 49}{2}$$ $$A = \frac{154}{2} = 77 \text{ cm}^2$$
Final answer: 77 cm².
Example 4
The area of a semicircle is $100\pi$ cm². Find its radius.
Start from the area formula and solve for $r$:
$$\frac{\pi r^2}{2} = 100\pi$$ $$\pi r^2 = 200\pi$$ $$r^2 = 200$$ $$r = \sqrt{200} = 10\sqrt{2} \approx 14.14 \text{ cm}$$
Final answer: $r = 10\sqrt{2} \approx 14.14$ 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. Use $\pi \approx 3.14$.
Sealing the full edge means the perimeter, both the arc and the flat base:
$$P = \pi r + 2r$$ $$P = 3.14 \times 0.5 + 2 \times 0.5$$ $$P = 1.57 + 1$$ $$P = 2.57 \text{ m}$$
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. Use $\pi \approx \dfrac{22}{7}$.
Two semicircles of equal radius make one full circle, so their combined area is just $\pi r^2$:
$$A = \pi r^2$$ $$A = \frac{22}{7} \times 35^2$$ $$A = \frac{22}{7} \times 1225$$ $$A = 3850 \text{ m}^2$$
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 ("discorectangle") 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 $P = \pi r$ and ignore the straight edge.
The correct way: Add the diameter: $P = \pi r + 2r$. 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 $\frac{\pi r^2}{2}$.
The correct way: Halve the diameter first: $r = d/2$. The memorizer who plugs numbers without checking which quantity the symbol stands for quadruples the area.
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). A quick units check at the end catches most of these.
Conclusion
A semicircle is half a circle, cut along the diameter.
Its area is $\dfrac{\pi r^2}{2}$ — half the circle's area.
Its perimeter is $\pi r + 2r$, 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.
Practice And A Next Step
Work through the exercises above and, for each one, decide first whether the question asks for area or perimeter — that single choice determines whether the diameter is part of your answer. Test your understanding with these problems before moving on.
Want a live Bhanzu trainer to walk through more semicircle problems? Book a free demo class.
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