How to Find the Area of a Semicircle

Half a circle, half the area. There is nothing more to the area than that, and if this page only had to cover it, it would be three lines long. The reason it is not is that a semicircle has a straight side, and that flat edge changes everything about measuring round the outside.

The area is the easy half

Take πr², halve it, done. For a radius of 10 the full circle holds 314.16 and the semicircle 157.08. If you were handed a diameter of 20 instead, you can halve it to get the radius or go straight there with π d² over 8, which is π times 400 over 8, landing on the same 157.08.

A semicircle is also just a sector with an angle of 180 degrees, so the sector formula works on it unchanged: 314.16 times 180 over 360. Three routes, one answer, and none of them needs anything you did not already have.

The perimeter is not half the circumference

Cut a circle in half and you create a new edge that was not there before. Walk round a semicircle and you follow the curve, then you have to come back along the flat. So the perimeter is the half circumference plus the diameter, not the half circumference on its own.

For our semicircle the arc is πr, which is 31.42, and the diameter is 20, so the perimeter is 51.42. Quoting 31.42 instead is 39 percent short. If that number is going to buy edging for a flower bed, trim for an arched window or fencing round a bay, the shortfall is a trip back to the supplier.

A worked check you can repeat on paper

The last two rows are for anyone using this shape structurally rather than in a homework question. The centre of area of a semicircle sits at 4r over 3π from the flat edge, which for a radius of 10 is 4.244 and not the 5 that instinct suggests. Semicircular sections are bottom heavy in that sense, and any calculation involving a first moment of area needs the 4.244.

Where the difference bites

There is a neat way to remember which halves and which does not. Area is a property of the inside, and cutting a circle in half genuinely halves the inside. Perimeter is a property of the boundary, and cutting a circle in half does not halve the boundary, it halves the curve and then adds a brand new straight piece that did not exist before.

Mistakes that catch people out

• Giving the perimeter as πr. The single most common error on this shape. It leaves out the diameter entirely, which here is 20 units of real boundary.
• Halving the diameter twice. Being handed 20 and using 10 as the diameter as well gives an area of 39.27, a quarter of the truth.
• Halving the radius instead of the area. The radius is not halved anywhere in this. Only the finished area is.
• Putting the centroid at half the radius. It sits at 4r over 3π, which is 0.4244 of the radius, not 0.5. Only matters for structural work, but there it matters a great deal.
• Forgetting that a quarter circle behaves the same way. Its perimeter is the arc plus two radii, not the arc alone, for exactly the same reason.

Try these

Common questions

What is the formula for the area of a semicircle?

π r² over 2, or π d² over 8 if you have the diameter. A radius of 10 gives 157.08 square units.

What is the perimeter of a semicircle?

The arc plus the diameter, so πr plus 2r. For a radius of 10 that is 31.42 plus 20, which is 51.42. It is not half the circumference, because cutting the circle created a straight edge that has to be walked as well.

How do you find the radius from the area?

Double the area, divide by π, then take the square root. An area of 157.08 doubles to 314.16, over π is 100, and the square root is 10.

Is a semicircle the same as a 180 degree sector?

Yes. The sector formula with an angle of 180 gives 314.16 times 180 over 360, which is the same 157.08. The semicircle just gets its own name because it turns up so often.

Where is the centroid of a semicircle?

At 4r over 3π from the straight edge, along the line of symmetry. For a radius of 10 that is 4.244, noticeably less than half the radius, which matters in any structural calculation using a first moment of area.

What about a quarter circle?

The area is π r² over 4, which for a radius of 10 is 78.54. The perimeter is the arc plus two radii rather than a diameter, because a quarter circle has two straight edges instead of one.

Related lessons and calculators

• Area of a sector, the general case this shape is the 180 degree version of.
• Circumference of a circle, where the arc of 31.42 comes from before the diameter is added.
• Area formulas for every common shape, including the sector, the segment and the ring.
• Circle arc calculator, for arcs that are not neat halves.
• Area of 2D shapes calculator, for running these without the arithmetic.
• the National Council of Teachers of Mathematics, for how circle geometry is sequenced in the curriculum.