Enter three side lengths to compute circumradius (R) and inradius (r) of any triangle
Side a
Side b
Side c
Result
Area K
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Circumradius R
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Inradius r
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R >= 2r:
Detailed Derivation
Circumradius & Inradius Formulas
s = (a + b + c) / 2
K = sqrt(s(s-a)(s-b)(s-c)) - Heron Formula
Circumradius R = abc / (4K)
Inradius r = K / s
Euler Inequality: R >= 2r
The circumradius is the radius of the circle passing through all three vertices. The inradius is the radius of the circle tangent to all three sides. Both are fundamental triangle measurements.
⚠Side lengths must satisfy the triangle inequality (a + b > c, a + c > b, b + c > a). Otherwise the triangle is degenerate or invalid.
What Are Circumradius and Inradius?
The circumradius R is the radius of the circumscribed circle that passes through all three vertices. The inradius r is the radius of the inscribed circle that touches all three sides. Together they describe the triangle's relationship with its circles.
Circumradius R
Radius of circumscribed circle. Larger for obtuse triangles. For right triangles, R = hypotenuse/2.
Inradius r
Radius of inscribed circle. Larger for more rounded triangles. Equilateral triangles have the maximum inradius for given perimeter.
Heron Formula
Area K = sqrt(s(s-a)(s-b)(s-c)). This is the bridge from side lengths to both R and r.
Euler Inequality
R >= 2r, with equality only for equilateral triangles. The distance d^2 = R(R - 2r).
Teaching Example: Triangle sides a=5, b=6, c=7. s = (5+6+7)/2 = 9. K = sqrt(9x4x3x2) = sqrt(216) = 14.697. R = (5x6x7)/(4x14.697) = 210/58.788 = 3.572. r = 14.697/9 = 1.633. R >= 2r: 3.572 >= 3.266 ok.
Circumradius R is the radius of the circle through all three vertices. Formula: R = abc/(4K). For right triangles, R = hypotenuse/2.
What is inradius?▼
Inradius r is the radius of the circle tangent to all three sides. Formula: r = K/s where s is the semiperimeter. The center is called the incenter.
What is the relationship between R and r?▼
Euler inequality: R >= 2r. Equality holds only for equilateral triangles. The distance between centers squared is d^2 = R(R - 2r).
How are R and r used in real life?▼
They are essential in engineering (gear design), architecture (arches and domes), navigation (celestial triangulation), and manufacturing (circular fittings).
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