The semicircle formula you need most often is area = (pi x r squared) / 2, where r is the radius. A semicircle is half a circle, so its area is half the area of the full circle. This guide gives you every common semicircle formula, including arc length and perimeter, worked examples, tips for using a diameter instead of a radius and a quick reference table.
Key takeaways
- Area of a semicircle = (pi x r squared) / 2, or pi x d squared / 8 using the diameter.
- Arc length of a semicircle = pi x r.
- Perimeter of a semicircle = pi x r + 2r, because you add the straight edge.
- Always check units, and square the radius, not the whole fraction.
- Use 3.14, 22/7 or your calculator’s pi key, depending on the accuracy you need.
What is a semicircle?
A semicircle is half of a circle, cut along a diameter. It has a curved edge (the arc) and a straight edge (the diameter). Its shape appears in arches, windows, protractors, garden beds and many geometry problems. For a quick definition and some history, see the Wikipedia page on the semicircle.
The main semicircle formula list
| Measurement | Using radius (r) | Using diameter (d) |
|---|---|---|
| Area | pi x r squared / 2 | pi x d squared / 8 |
| Curved arc length | pi x r | pi x d / 2 |
| Straight edge | 2r | d |
| Perimeter | pi x r + 2r = r(pi + 2) | pi x d / 2 + d |
If you only remember one thing, remember that everything comes from the full circle: circumference is 2 x pi x r and area is pi x r squared, and you then halve them.
How to find the area step by step
- Find the radius. If you know the diameter, divide it by 2.
- Square the radius (multiply it by itself).
- Multiply by pi.
- Divide by 2.
- Write the answer in square units, such as cm squared.
Example 1: radius of 6 cm
Radius squared is 36. Multiply by pi to get about 113.1. Half of that is about 56.5. The area is roughly 56.5 cm squared.
Example 2: diameter of 10 m
The radius is 5 m. Radius squared is 25. Times pi is about 78.54. Half is about 39.27. The area is about 39.3 m squared. Using the diameter version, pi x 100 / 8 gives the same result.
How to find the perimeter and arc length
The arc is half the circumference of the full circle, so it equals pi x r. The perimeter of a semicircle includes the straight diameter as well, so you add 2r. This is where many students slip: if a question asks for the perimeter, do not forget the straight edge.
Example 3: radius of 4 inches
Arc length = pi x 4, which is about 12.57 inches. Diameter = 8 inches. Perimeter = 12.57 + 8 = 20.57 inches.
Example 4: finding the radius from the perimeter
If the perimeter is 30.84 cm, use P = r(pi + 2). Then r = 30.84 / 5.14, which is about 6. The radius is 6 cm.
Working backward
You can rearrange the semicircle formula to find the radius from the area: r = the square root of (2A / pi). For instance, if the area is 50 square units, 2A is 100, divided by pi is about 31.83, and the square root is about 5.64. So the radius is about 5.64 units.
Common mistakes to avoid
- Squaring the wrong thing: In pi x r squared / 2, only the radius is squared.
- Mixing up radius and diameter: If you use the diameter as the radius, your area will be four times too large.
- Forgetting the straight edge: Perimeter includes the diameter, but arc length does not.
- Mismatched units: Convert everything to the same unit before calculating.
- Early rounding: Keep extra decimal places until the final step.
Real-life uses
- Calculating the glass or paint needed for a semicircular window.
- Estimating the area of a half-round garden bed or patio.
- Working out the length of edging around a half-circle lawn.
- Combining shapes: a rectangle with a semicircular end, such as a tunnel cross-section or a running-track end, uses a rectangle’s area plus a semicircle’s area.
Composite shape example
A rectangle 8 m by 4 m has a semicircle attached on one 4 m side. The semicircle’s diameter is 4 m, so its radius is 2 m. Its area is pi x 4 / 2, about 6.28 m squared. The rectangle’s area is 32 m squared. The total area is about 38.28 m squared.
A note on the centroid
For more advanced courses, the centroid of a solid semicircle sits along its axis of symmetry at a distance of 4r / (3 pi) from the straight edge. This is used in physics and engineering for balance and stress calculations.
If you are studying related math topics, our explainers on the four quadrants of a graph, what a quotient is and the C2H4 Lewis structure are good companions.
Practice problems
Try these quick questions to test the semicircle formula. First, find the area of a semicircle with a radius of 10 cm; the answer is about 157.1 cm squared. Second, find the perimeter of a semicircle with a diameter of 14 m, using pi as 22/7; the arc is 22 m, plus the 14 m diameter, giving 36 m. Third, a semicircular window has a radius of 0.5 m and you want to know how much trim goes around the curved edge: that is pi x 0.5, about 1.57 m. Work through each step on paper, write the units and check that your answer is reasonable by comparing it with the full circle. A semicircle’s area should always be exactly half of the circle’s.
Frequently asked questions
What is the area of a semicircle formula?
Area = (pi x r squared) / 2, where r is the radius.
How do I find the area from the diameter?
Use pi x d squared / 8, or halve the diameter to get the radius and use the radius formula.
What is the perimeter of a semicircle?
Perimeter = pi x r + 2r, which is the arc plus the straight diameter.
Is arc length different from perimeter?
Yes. Arc length is only the curved part (pi x r). Perimeter also includes the diameter.
Which value of pi should I use?
Use your calculator’s pi button for accuracy, or 3.14 or 22/7 if your teacher or textbook asks for it.
Conclusion
Once you remember that the semicircle formula is half of the circle formula, everything else follows. Use area = pi x r squared / 2, arc length = pi x r and perimeter = pi x r + 2r, and double check whether you have the radius or diameter. Practice with a few examples and these calculations become quick.
Key takeaways about semicircle formula
Here is a short summary of semicircle formula so you can find the main points quickly.
- The semicircle formula you need most often is area = (pi x r squared) / 2, where r is the radius.
- A semicircle is half of a circle, cut along a diameter.
- If you only remember one thing, remember that everything comes from the full circle: circumference is 2 x pi x r and area is pi x r squared, and you then halve them.
- The arc is half the circumference of the full circle, so it equals pi x r.
Good decisions about semicircle formula come from checking the facts, comparing your options and taking your time.
Related reading:
- C2H4 Skeletal Structure and Lewis Structure Explained
- Rules of the Road USCG Test: What to Study and How
For general background, you can look up semicircle formula on Wikipedia.
