Semicircle Area Calculator

What is a Semicircle?

A semicircle is a one-dimensional locus of points that forms half of a circle. It is created when a full circle is divided into two equal halves along any straight line passing through its center (a diameter line). The word originates from the Latin prefix semi-, meaning half or partly.

Geometrically, a closed semicircle consists of two distinct boundaries: a curved circular arc spanning exactly 180 degrees ($\pi$ radians) and a straight flat edge that coincides with the circle diameter. Calculate other circular and curved geometric properties with our Circle Calculator, Circle Segment Calculator, and Arc Length Calculator.

Semicircle Area Formula

Because a semicircle is exactly half of a complete circle, its surface area is half of the area of a circle with the same radius:

$$A = \frac{1}{2} \pi r^2 = \frac{\pi d^2}{8}$$

Where:

A is the surface area of the semicircle
r is the radius (distance from the center of the base to any point on the arc)
d is the diameter of the base, equal to $2r$
π is the mathematical constant Pi (≈ 3.14159265)

Semicircle Perimeter vs Arc Length

A common point of confusion is the difference between the curved arc length and the total perimeter of a semicircle:

Curved Arc Length ($L$): The curved path along the perimeter, equal to half of the full circle circumference: $$L = \pi r$$
Total Perimeter ($P$): The entire closed boundary, which includes both the curved arc and the straight diameter base: $$P = \pi r + d = \pi r + 2r = r(\pi + 2) \approx 5.14159 \times r$$

Centroid of a Semicircle

The geometric center of mass (centroid) of a uniform planar semicircle lies along the vertical axis of symmetry at a distance of $y_c$ from the flat straight edge:

$$y_c = \frac{4r}{3\pi} \approx 0.4244 \times r$$

This formula is essential in structural engineering, mechanics of materials, and civil design when computing moments of inertia, centroidal axes, and bending stresses for semi-circular arches or conduits.

How to Calculate Semicircle Measurements

From Radius: Multiply the square of the radius by $\pi$ and divide by 2 for the area. Multiply the radius by $(\pi + 2)$ for the total perimeter.
From Diameter: Divide diameter by 2 to obtain the radius, then apply the area and perimeter formulas.
From Area: Reorganize the area formula to solve for radius: $r = \sqrt{2A / \pi}$. Once radius is known, calculate perimeter and diameter.
From Perimeter: Compute radius using $r = P / (\pi + 2)$, then calculate the corresponding area.

Frequently Asked Questions

Is the perimeter of a semicircle half of a full circle circumference?

No. The perimeter of a semicircle is strictly greater than half the circumference of a circle. While the curved edge is half of the circumference (πr), a closed semicircle also includes the flat diameter base (2r). Therefore, the total perimeter is πr + 2r = r(π + 2).

What is the area of a semicircle with a radius of 10 cm?

Using the formula A = 0.5 × π × r², with r = 10 cm: A = 0.5 × π × 100 = 50π ≈ 157.0796 cm².

How do you find the radius of a semicircle if only the area is given?

Multiply the area by 2, divide the result by π, and take the square root: r = √(2A / π). For instance, if the area is 50 cm², r = √(100 / π) ≈ 5.6419 cm.

Where is the center of gravity of a semicircle located?

The center of mass (centroid) is located on the axis of symmetry at a distance of 4r / (3π) ≈ 0.4244 × r measured perpendicularly upward from the flat base.

What is Thales's Theorem regarding semicircles?

Thales's Theorem states that any angle inscribed in a semicircle where the hypotenuse is the diameter of the circle is always a right angle (90 degrees). Any point chosen on the semicircle arc connected to both endpoints of the diameter forms a right-angled triangle.

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