Semicircle Formula Function: Area, Perimeter, Arc Length

The semicircle formula function you need most often is the area: A = (πr²) / 2, where r is the radius. The perimeter is P = πr + 2r, which adds the curved edge to the straight diameter. A semicircle is simply half of a circle, so each formula comes from halving the matching full-circle formula, with one important exception for the perimeter.

The key semicircle formulas

Quantity	Using radius (r)	Using diameter (d)
Area	πr² / 2	πd² / 8
Arc length (curved edge)	πr	πd / 2
Straight edge (diameter)	2r	d
Perimeter	πr + 2r = r(π + 2)	πd / 2 + d = d(π + 2) / 2

Here π (pi) is the constant approximately equal to 3.14159. For many school problems you can use 3.14 or the π button on your calculator, and your teacher or exam paper will usually say which to use. The background on the constant itself is on Wikipedia’s article on pi.

Where the formulas come from

A full circle with radius r has an area of πr² and a circumference of 2πr. A semicircle is exactly half of that disc, so its area is half of πr², giving πr² / 2. The curved part of its boundary is half of the circumference, which is πr.

The perimeter needs care. A semicircle is a closed shape, so its boundary has two parts: the curved arc and the straight diameter. That is why the perimeter is not just half the circumference. You must add the diameter, 2r, which gives P = πr + 2r. Forgetting this straight edge is one of the most common errors.

Worked example 1: area from the radius

A semicircular garden bed has a radius of 4 m. Find its area.

1. Write the formula: A = πr² / 2.
2. Square the radius: 4² = 16.
3. Multiply by π: 16 × 3.14159 ≈ 50.27.
4. Halve it: 50.27 / 2 ≈ 25.13.

The area is about 25.13 square metres.

Worked example 2: perimeter from the diameter

A semicircular window has a diameter of 1.2 m. Find the perimeter of the frame, including the straight bottom.

1. Find the radius: r = 1.2 / 2 = 0.6 m.
2. Arc length: πr = 3.14159 × 0.6 ≈ 1.885 m.
3. Add the diameter: 1.885 + 1.2 = 3.085 m.

The perimeter is about 3.09 m. If you were fitting trim only on the curved edge, you would use the arc length (about 1.89 m) and not the full perimeter.

Worked example 3: finding the radius from the area

Sometimes you know the area and need the radius. Rearrange A = πr² / 2:

1. Multiply both sides by 2: 2A = πr².
2. Divide by π: r² = 2A / π.
3. Take the square root: r = √(2A / π).

For an area of 50 cm², r = √(100 / 3.14159) = √31.83 ≈ 5.64 cm. The diameter is then about 11.28 cm.

Using the formulas in a spreadsheet or calculator

If you want to treat the semicircle formula as a function in a spreadsheet, you can enter the radius in a cell and use the built-in value of pi. For a radius in cell A1, the formulas are as follows.

• Area: =PI()*A1^2/2
• Arc length: =PI()*A1
• Perimeter: =PI()*A1+2*A1

These work in most major spreadsheet programs, including Excel, Google Sheets and LibreOffice Calc. In a programming language, you would define a function that takes the radius and returns the area, for example by multiplying the radius squared by pi and dividing by two. Use the language’s built-in pi constant rather than typing 3.14, because it gives better accuracy.

Common mistakes to avoid

• Forgetting to halve. The area of a full circle is πr², so make sure you divide by 2 for a semicircle.
• Leaving out the diameter in the perimeter. The boundary includes the straight edge.
• Using the diameter instead of the radius. If the question gives you the diameter, halve it first, or use the diameter formulas.
• Squaring incorrectly. Square only the radius, not π × r. The expression πr² means π × (r × r).
• Mixing units. Do not combine centimetres and metres, and remember that area is in square units.
• Rounding too early. Keep extra decimal places during the calculation and round at the end.

Real-world uses

Semicircles appear in arched windows and doorways, running tracks, tunnels, garden beds, patios and roof details. Builders and designers use the area to estimate paint, paving or glass and the perimeter to estimate edging, trim or fencing. In cooking and craft, you might need to cut a half-circle of dough or fabric. As an everyday example, the same geometry that helps you estimate a semicircular pizza or pie portion is the basis for working out a round pan in the kitchen, as discussed in our deep fryer oil temp guide, where the size of the pan affects how much oil you need.

If you are working with measurements in imperial units, our short guide to the inch designation and the mark for inches explains how to write them properly. And if you are a gamer who enjoys the maths behind city layouts, you may like our guide to the best civilization in Rise of Kingdoms.

Frequently asked questions

Conclusion

Once you know the semicircle formula function for area, arc length and perimeter, most problems come down to four steps: find the radius, apply the right formula, include the straight edge when you need the perimeter, and keep your units consistent. Practise a few examples and the formulas will stick.