How To Calculate the Radius of the Circle Inscribed in a Triangle

• How to find the center of the circle inscribed in a triangle?

• In any triangle the bisectors of the angles cross at the same point, which is the center of the circle tangent to the sides of the triangle, called the circle inscribed in the triangle.
• 

  The bisectors and the circle inscribed in the triangle

Bisectors AA', BB' and CC' intersect at point O, which is the center of the circle inscribed in the triangle. OP ⊥ AB, OM ⊥ BC, ON ⊥ AC and OM = ON = OP = r, where r is the radius of the circle inscribed in the triangle.

» How To Find the Center of the Circle Inscribed in a Triangle

• How to calculate the radius of the inscribed circle

• To calculate the radius of the circle inscribed in the triangle, divide the area of ​​the triangle by its semiperimeter:

• r = ^Area / _s
• Where:
• r = radius of the circle inscribed in the triangle
• Area = area of the triangle
• s = the semiperimeter of the triangle

• Why this formula? Let's have a closer look

• In the triangle △ABC, where AA', BB' and CC' are the bisectors of the angles ∠BAC, ∠ABC and ∠ACB, respectively, we calculate its area as the sum of the three smaller triangles, △OAB, △OBC and △OAC:
• 

  The bisectors and the circle inscribed in the triangle

Area = s × r    =>    r = ^Area / _s

• An example of the radius of the inscribed circle calculation

• Let's calculate the radius of the circle inscribed in a triangle whose area is equal to 113.137084989848 square units and whose semiperimeter is equal to 32 units.