Circumradius

This formula gives the radius of the circle that passes through all three vertices of a triangle. It works because a triangle’s side lengths and area determine its size uniquely, and the circumradius can be derived by combining the area formula with the sine rule: the area is linked to two sides and the included angle, while the extended sine rule connects the sides to the same angle through the circle’s radius. Putting those relationships together eliminates the angles and leaves R in terms of a, b, c, and K.

Use this when you know all three side lengths of a triangle and its area, and you want the radius of the triangle’s circumcircle. It is especially useful in advanced geometry, trigonometry, and problems involving cyclic quadrilaterals or triangle construction.

• Using the semiperimeter instead of the area in the denominator
• Forgetting that K must be the triangle’s area, not the area of the circumcircle
• Mixing up the circumradius with the inradius, which uses a different formula