Semi-circle
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A semicircle of radius r.

In mathematics (more specifically geometry), a semicircle is a two-dimensional geometric shape that forms half of a circle. Being half of a circle's 360°, the arc of a semicircle always measures 180°. A triangle inscribed in a semicircle is always a right triangle.

Uses

A semicircle can be used to construct arithmetic and geometric means of two lengths using straight-edge and compass. If we make a semicircle with a diameter of a+b, then the length its radius is the arithmetic mean (since it's half of the diameter). The geometric mean can be found by dividing the diameter into two segments of lengths a and b, and then connecting their common end and the semicircle with a segment perpendicular to the diameter. The length of the resulting segment is the geometric mean¹, which can be proved using Pythagorean theorem.

This method can be used to accomplish quadrature of a rectangle (since a square whose sides are equal to geometric mean of sides of a rectangle has the same area as the rectangle), and thus any figure for which we can construct a rectangle with equal area, such as any polygon (but not a circle).

References

See also

• Thales' theorem

External links

• Semicircle - Mathworld