Area of a Semi-Circle Calculator

What a semicircle is

A semicircle is exactly half a circle, cut along its diameter. It has one curved edge (the arc, which is half the full circle’s circumference) and one straight edge (the diameter). Because it is half a circle, its area is always half the area of the full circle with the same radius.

The formulas, explained plainly

Area: A = pi x r^2 / 2. Take the ordinary circle area and halve it, because you only have half the circle.

Arc length: the curved edge alone is pi x r, which is half of the full circumference 2 x pi x r.

Perimeter: P = pi x r + 2r. This is the whole way around the shape: the curved arc plus the straight diameter edge.

Working backwards: if you know the area, r = sqrt(2 x area / pi). If you know the perimeter, r = perimeter / (pi + 2). If you know the arc length, r = arc / pi. If you know the diameter, halve it first. The calculator above does all of these in one step and shows the arithmetic.

The perimeter trap

Halving the circumference gives you only the curved part. It leaves out the diameter, which is a real edge of the shape, and it understates the perimeter by about 39 percent. This is the mistake that costs marks in exams and material in workshops. The check is simple: ask whether the boundary you are measuring includes the flat side. If you are edging a semicircular garden bed against a wall, the wall is the diameter and you only need the arc. If the bed stands free, you need both.

A related slip: squaring the diameter and then halving gives four times too much area. Always halve the diameter to get the radius first, then square.

Worked example

A semicircular window has a radius of 7 cm.

Area = pi x 7^2 / 2 = 3.14159 x 49 / 2 = 76.97 cm^2
Arc = pi x 7 = 21.99 cm
Perimeter = 21.99 + 14 = 35.99 cm

Enter 7 as the radius in the calculator above and you will get the same numbers, plus the diameter (14 cm) and the full-circle area (153.94 cm^2) for reference. Working backwards: enter 35.99 as the perimeter and the calculator recovers a radius of 7 cm exactly.

Frequently asked questions

What is the difference between the arc and the perimeter of a semicircle?

The arc is only the curved edge: pi x r. The perimeter is the whole boundary: the arc plus the straight diameter, pi x r + 2r. For a radius of 5, the arc is 15.71 and the perimeter is 25.71.

How do I find the area if I only know the diameter?

Halve the diameter to get the radius, then use A = pi x r^2 / 2. Or select “Diameter” in the calculator and enter it directly. A diameter of 10 gives a radius of 5 and an area of about 39.27 square units.

Can I solve for the radius if I know the area?

Yes: r = sqrt(2 x area / pi). Select “Area” in the calculator. An area of 76.97 recovers a radius of 7.

What units should I use?

Any unit works as long as you are consistent: centimeters, meters, inches, or feet in give square centimeters, square meters, square inches, or square feet out. The calculator labels results in generic units so the math works for all of them.

Is a semicircle the same as a half circle?

Yes. Semicircle is the geometry term for a half circle: a 180-degree sector of a circle, bounded by a diameter and the arc between its endpoints.

Why is the full circle area shown too?

As a built-in sanity check. The semicircle area should always be exactly half of it. If it is not, something went wrong with the input.

Limitations to know

This calculator assumes a perfect semicircle: a true 180-degree half circle with a straight diameter. It does not handle elliptical halves, circular segments cut at other angles, or three-dimensional half spheres. Results are rounded to six decimals, which is more than enough for homework, construction estimates, and design work.