Semicircle Area Calculator
Calculate area, perimeter, and arc length of any semicircle from radius, diameter, or area.
📊 What is a Semicircle?
A semicircle is exactly half of a circle, formed by cutting a complete circle along its diameter. The boundary of a semicircle consists of two parts: the curved arc (which is exactly half the circumference of the full circle) and the straight diameter (the flat edge created by the cut). Together, these form the complete perimeter of the semicircle.
The semicircle is one of the most commonly encountered geometric shapes in the real world. Arched windows and doorways, tunnel cross-sections, the D-shaped area in football and basketball courts, protractors, and half-round mouldings are all semicircular. Understanding how to calculate the area and perimeter of a semicircle is essential knowledge in architecture, civil engineering, product design, and everyday practical geometry.
The area of a semicircle is always exactly half the area of the full circle with the same radius. Since a full circle has area pi times r squared, the semicircle area is pi times r squared divided by 2. The arc length (curved portion) is exactly half the circumference of the full circle, giving pi times r. The total perimeter includes both the arc and the straight diameter, totalling r times (pi + 2), which is approximately 5.14159 times the radius.
What makes the semicircle particularly useful to work with is that you only need one measurement to calculate everything else. If you know the radius, diameter, or even just the area, this calculator derives all other properties instantly. The relationships between radius, diameter, arc, area, and perimeter are all direct consequences of the circle formulas divided by two, making them elegant and straightforward to remember.
In engineering applications, the centroid (centre of mass) of a semicircular area is located at a distance of 4r/(3pi) from the flat diameter, approximately 0.4244 times the radius. This is important for structural calculations involving semicircular cross-sections in beams, plates, and arches.
📐 Formula and Derivation
All semicircle formulas are derived from the corresponding full circle formulas, divided by 2 where applicable.
Given radius r:
Given diameter d: r = d / 2, then apply the radius formulas above.
Given area A:
📖 How to Use This Calculator
Steps to Calculate
💡 Example Calculations
Example 1 - Radius of 7 units
A semicircular window has a radius of 7 cm. Find its area and frame length (perimeter).
Example 2 - Diameter of 10 units
A semicircular tunnel has a diameter of 10 m. Find the cross-sectional area and curved tunnel lining length.
Example 3 - Area of 25 square units
A garden bed has a semicircular area of 25 sq m. Find the radius and the length of edging needed.
Example 4 - Radius of 3 units
A small semicircular gasket has a radius of 3 mm. Find all dimensions for the manufacturing specification.
Frequently Asked Questions
🔗 Related Calculators
What is the area of a semicircle?
The area of a semicircle is A = pi times r squared divided by 2, where r is the radius. It is exactly half the area of a full circle. For a semicircle with radius 7 cm, Area = 3.14159 times 49 / 2 = 76.969 sq cm. If you know the diameter d, the radius is r = d / 2.
What is the perimeter of a semicircle?
The perimeter (total boundary length) of a semicircle = arc length + diameter = pi times r + 2r = r(pi + 2). For radius 7, perimeter = 7 times (3.14159 + 2) = 7 times 5.14159 = 35.991 cm. The perimeter includes both the curved arc and the straight flat edge.
What is the arc length of a semicircle?
The arc length (curved edge only) of a semicircle = pi times r, which is half the circumference of the full circle. For radius 5 cm, arc = 3.14159 times 5 = 15.708 cm. Note that the arc length does not include the straight diameter edge.
How do I find the radius from the area of a semicircle?
Rearrange A = pi r squared / 2 to get r = sqrt(2A / pi). For example, if area = 25 sq cm, r = sqrt(2 times 25 / 3.14159) = sqrt(50 / 3.14159) = sqrt(15.915) = 3.989 cm approximately 4 cm.
What is the formula for the diameter of a semicircle?
The diameter is simply twice the radius: d = 2r. The diameter forms the straight flat edge of the semicircle. If you know the area A, then d = 2 times sqrt(2A / pi).
What is the difference between a semicircle and a half circle?
A semicircle and a half circle are the same thing - a semicircle is literally half of a circle, formed by cutting a full circle along its diameter. The semicircle shape includes both the curved arc (half circumference) and the straight diameter edge.
How is the semicircle perimeter different from the arc length?
The arc length is only the curved part of the semicircle boundary: arc = pi times r. The full perimeter includes both the arc and the straight diameter: perimeter = pi times r + 2r = r(pi + 2). For radius 5, arc = 15.708, perimeter = 25.708.
What are practical uses of semicircle area calculations?
Semicircle area calculations appear in architecture (arched windows and doorways), civil engineering (tunnel cross-sections), sports (semicircular playing areas like the D in football), and manufacturing (half-round gaskets and seals). Any curved structure that is half a circle requires these formulas.
How does the semicircle area relate to the full circle area?
The area of a semicircle is always exactly half the area of the full circle with the same radius. Full circle area = pi r squared. Semicircle area = pi r squared / 2. Similarly, the arc of a semicircle is half the circumference of the full circle.
What is a semicircle with radius 3?
For radius r = 3: Area = pi times 9 / 2 = 14.137 sq units. Arc length = pi times 3 = 9.425 units. Perimeter = 3 times (pi + 2) = 3 times 5.14159 = 15.425 units. Diameter = 6 units.