An online calculator to calculate the radius R of a circumscribed circle of a triangle of sides a, b and c.
Example
Use the two formulas given above to find the radius of the circumscribed circle to the triangle with sides 6, 7 and 10 cm.
Solution
1) We use the first formula \( 2 R = \dfrac{a}{\sin(A)} \) by first using the cosine law to find angle A
\( a^2 = b^2 + c ^2 - 2 b c cos(A)) \)
\( cos(A) = \dfrac{a^2 - b^2 - c ^2}{-2 b c} \)
A = arccos \( \dfrac{a^2 - b^2 - c ^2}{-2 b c} \)
= arccos \( \dfrac{6^2 - 7^2 - 10 ^2}{-2 (7) (10) } = 36.18^{\circ} \)
R = \( 0.5 \dfrac{6}{\sin(36.18)} = 5.08\)
2) We now use the second formula
\( s = 0.5(a + b + c) = 0.5(6 + 7 + 10) = 11.5 \)
\( R = \dfrac{a \cdot b \cdot c}{ 4 \sqrt{s(s-a)(s-b)(s-c)}} = \dfrac{6 \cdot 7 \cdot 10}{4 \sqrt{11.5(11.5-6)(11.5-7)(11.5-10)}} = 5.08 \)
Use the calculator to check the results of the above example.