Semicircle Calculator
Calculate the area, perimeter, and arc length of a semicircle instantly from its radius or diameter. Enter your measurement in the panel on the left, select your unit, and get a complete breakdown displayed as a chat bubble result. Use the AI chat below for follow-up questions about geometry, real-world applications, or unit conversions.
What Is a Semicircle?
A semicircle is exactly half of a full circle, created by cutting a circle along its diameter. It has one straight edge (the diameter) and one curved edge (half the circumference, also called the arc). The semicircle is one of the most common shapes in architecture, engineering, and everyday design because it combines the structural strength of an arch with the visual elegance of a curve.
A semicircle has several key properties: the straight edge always passes through the center of the original circle (making it the diameter), all points on the curved edge are equidistant from the center, and any angle inscribed in a semicircle that touches both ends of the diameter is always exactly 90 degrees (Thales' theorem). Real-world semicircles include arched doorways and windows, half-pipe skateboard ramps, protractors, stadium end zones, and the cross-sections of tunnels.
How the Semicircle Calculator Works
Calculating from Radius
The radius is the distance from the center of the semicircle (midpoint of the diameter) to any point on the curved edge. When you select "Radius" and enter a value, the calculator applies three formulas simultaneously: Area = πr²/2, Perimeter = r(π + 2), and Arc Length = πr. The result table shows all five key measurements — area, perimeter, arc length, radius, and diameter — in a single chat bubble.
Calculating from Diameter
The diameter of a semicircle is the length of the straight edge — the full width across the flat base. If you know the diameter (for example, you measured across an arched window opening), select "Diameter," enter the value, and the calculator converts it to radius (r = d/2) before applying all formulas. Measuring the diameter is often easier in practice because you can place a tape measure across the flat base without needing to find the center point.
Understanding the Results
The calculator returns three distinct measurements. The area is the total flat surface inside the semicircle — useful for material coverage, floor plans, and garden plots. The perimeter is the total distance around the entire boundary of the semicircle, including both the curved arc and the straight diameter — useful for fencing, framing, and border materials. The arc length is only the curved portion of the boundary — useful when you need to know how much trim, molding, or curved material to buy.
Semicircle Formulas
Area of a Semicircle (A = πr²/2)
The area of a semicircle is exactly half the area of the full circle with the same radius. Since the full circle area is πr², the semicircle area is πr²/2. For a semicircle with radius 5 cm: A = (3.14159 × 25) / 2 = 78.54 / 2 = 39.27 cm². This is the most commonly asked semicircle formula in math classes and is used in architecture to calculate the surface area of arched sections.
Perimeter of a Semicircle (P = πr + 2r)
The perimeter of a semicircle is the total distance around the entire shape: the curved arc (πr) plus the straight diameter (2r). For radius 5 cm: P = (3.14159 × 5) + (2 × 5) = 15.708 + 10 = 25.708 cm. Note that the perimeter is not simply half the circumference of the full circle — it includes the straight edge across the bottom. This distinction is important when calculating border materials for a semicircular design.
Arc Length of a Semicircle (L = πr)
The arc length is only the curved portion of the semicircle boundary — it does not include the diameter. The arc of a semicircle is exactly half the circumference of the full circle (C/2 = 2πr/2 = πr). For radius 5 cm: L = 3.14159 × 5 = 15.708 cm. This value is what you need when installing curved trim, bent piping, or any material that follows only the curved edge and not the straight base.
Semicircle vs Full Circle
The key difference between a semicircle and a full circle is that a semicircle adds the diameter as a straight edge. Here is a comparison of all formulas for a circle with radius r:
| Property | Full Circle | Semicircle |
|---|---|---|
| Area | πr² | πr²/2 |
| Boundary length | 2πr (circumference) | πr + 2r (arc + diameter) |
| Curved edge | 2πr | πr |
| Straight edge | None | 2r (diameter) |
| For r = 5 cm: Area | 78.54 cm² | 39.27 cm² |
| For r = 5 cm: Boundary | 31.42 cm | 25.71 cm |
Note that the semicircle's perimeter (25.71 cm) is not half the circle's circumference (31.42 cm). It is less than half because the diameter (10 cm) is shorter than the arc alone (15.71 cm). This is a common source of confusion when students first work with semicircle problems. You can also use our circumference calculator to verify the full circle values.
How to Find the Area of a Half Circle
Finding the area of a half circle follows three simple steps:
- Identify the radius. If you have the diameter, halve it. If you have the arc length, divide by π to get the radius.
- Square the radius. Multiply the radius by itself: r × r = r².
- Apply the formula. Multiply r² by π (≈ 3.14159), then divide by 2: A = πr²/2.
Example: Semicircle with radius 8 m
Step 1: r = 8 m (given)
Step 2: r² = 64 m²
Step 3: A = (3.14159 × 64) / 2 = 201.06 / 2 = 100.53 m²
You can also reverse this process to find the radius from the area: r = √(2A/π). For a semicircular garden with an area of 50 m²: r = √(100/3.14159) = √31.83 = 5.64 m. The calculator handles both directions — enter a radius or diameter to get the area instantly, then ask the AI to find the radius from a given area if needed.
Practical Uses of Semicircle Calculations
Semicircle geometry comes up in many real-world fields:
- Architecture — arched windows and doorways — Semicircular arches are among the oldest architectural elements. A contractor fitting an arched window needs the arc length to cut the curved trim and the area to calculate the glass pane size. A semicircular window with a 36-inch diameter needs a glass pane of πr²/2 = π × 18² / 2 ≈ 508.9 sq in and a trim piece of πr = π × 18 ≈ 56.5 inches.
- Landscaping — semicircular garden beds — Gardeners often design semicircular beds against a fence or wall. Knowing the area tells you how much soil, mulch, or ground cover to purchase. A semicircular bed with a 3-meter radius has an area of π × 9 / 2 ≈ 14.14 m² and a curved border of π × 3 ≈ 9.42 meters of edging.
- Engineering — pipe and tunnel cross-sections — Many drainage pipes, culverts, and tunnels have semicircular cross-sections. The area of the semicircle determines the flow capacity of the pipe, while the arc length is used to calculate the material needed to form the curved wall.
- Design — logos, UI elements, icons — Graphic designers and UI/UX designers frequently work with semicircles in icons, buttons, and illustrations. Knowing the exact arc length and dimensions ensures pixel-accurate design and consistent scaling.
- Sports — half-pipes and tracks — Skateboard half-pipes, BMX ramps, and running track end curves are semicircular. The radius determines the difficulty level of the ramp, and the arc length determines how much surface material (concrete, wood, or rubber) is needed.
Semicircle Calculator Examples
Example 1 — Semicircle with Radius 5 cm
Given: radius = 5 cm. Area = π × 25 / 2 = 39.27 cm². Perimeter = π × 5 + 2 × 5 = 15.708 + 10 = 25.708 cm. Arc length = π × 5 = 15.708 cm. Diameter = 10 cm. A semicircle of this size is roughly the shape of a standard protractor. The arc length (15.71 cm) is the measurement along the curved edge, while the perimeter (25.71 cm) includes the full boundary — useful if you were framing the shape in wood or metal.
Example 2 — Semicircular Window with Diameter 36 inches
Given: diameter = 36 inches, so radius = 18 inches. Area = π × 18² / 2 = π × 324 / 2 = 508.94 sq in (approximately 3.54 sq ft). Perimeter = π × 18 + 2 × 18 = 56.549 + 36 = 92.549 inches. Arc length = π × 18 = 56.549 inches (approximately 4.71 feet of curved trim). This is a common size for a Palladian-style arched window. A contractor would order 509 sq in of glass and 56.6 inches of curved wood molding for the arc, plus 36 inches of straight sill trim for the base.
Frequently Asked Questions
What is the area of a half circle?
The area of a half circle (semicircle) is A = πr²/2, where r is the radius. This is exactly half the area of a full circle with the same radius. For example, a semicircle with radius 10 cm has an area of π × 100 / 2 = 157.08 cm². If you know the diameter instead, use r = d/2 first, then apply the formula.
How do you find the perimeter of a semicircle?
The perimeter of a semicircle equals the arc length plus the diameter: P = πr + 2r = r(π + 2). You cannot simply halve the circumference of the full circle because that only gives you the arc (πr) — you must also add the straight diameter (2r) to get the full boundary. For radius 7 m: P = 7 × (3.14159 + 2) = 7 × 5.14159 = 35.991 m.
Is the perimeter of a semicircle half the circumference?
No. The perimeter of a semicircle is NOT half the circumference of the full circle. The perimeter includes the straight diameter, while half the circumference is only the arc length. For radius r: half circumference = πr, but semicircle perimeter = πr + 2r. For r = 10: half circumference = 31.42, but semicircle perimeter = 51.42. The perimeter is always greater than half the circumference.
What is the difference between arc length and perimeter?
The arc length is only the curved portion of the semicircle boundary: L = πr. The perimeter is the total boundary including both the arc AND the straight diameter: P = πr + 2r. The arc length is useful when you only need to measure or cut the curved edge (like curved molding or bent pipe). The perimeter is used when you need the total distance around the entire shape (like fencing or framing).
How do you find the radius from a semicircle area?
To find the radius from a semicircle area, reverse the area formula: r = √(2A/π). For example, if the area is 100 cm²: r = √(200/3.14159) = √63.66 = 7.979 cm. You can verify this: A = π × 7.979² / 2 = π × 63.66 / 2 = 199.97/2 ≈ 100 cm². This is useful in landscaping and architecture when you know the desired area and need to find the appropriate radius.