Inradius

The inradius r is the distance from the center of a triangle’s incircle to any side, so the incircle touches all three sides without crossing them. The key idea is that a triangle’s area can be split into three smaller triangles, each with height r and bases equal to the side lengths, which gives K = rs where s is the semiperimeter. Rearranging that area relationship yields r = K/s.

Use this when you know a triangle’s area and its side lengths, or when you can find the semiperimeter from the sides. It is especially useful in geometry proofs, optimization problems, and any problem involving an incircle or tangent lengths.

• Using the full perimeter instead of the semiperimeter s
• Forgetting that K must be the triangle’s area, not a side length or height
• Mixing up the inradius with the circumradius, which is the radius of the circle through the vertices