Triangle calculator SSS - result

Please enter the triangle sides:


Acute scalene triangle.

Sides: a = 12   b = 18   c = 21

Area: T = 107.7898856103
Perimeter: p = 51
Semiperimeter: s = 25.5

Angle ∠ A = α = 34.77219440319° = 34°46'19″ = 0.60768849107 rad
Angle ∠ B = β = 58.81113776665° = 58°48'41″ = 1.02664521779 rad
Angle ∠ C = γ = 86.41766783015° = 86°25' = 1.5088255565 rad

Height: ha = 17.96548093505
Height: hb = 11.9776539567
Height: hc = 10.26656053431

Median: ma = 18.6154510469
Median: mb = 14.54330395722
Median: mc = 11.12442977306

Inradius: r = 4.22770139648
Circumradius: R = 10.52105680902

Vertex coordinates: A[21; 0] B[0; 0] C[6.21442857143; 10.26656053431]
Centroid: CG[9.07114285714; 3.42218684477]
Coordinates of the circumscribed circle: U[10.5; 0.65875355056]
Coordinates of the inscribed circle: I[7.5; 4.22770139648]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.2288055968° = 145°13'41″ = 0.60768849107 rad
∠ B' = β' = 121.1898622333° = 121°11'19″ = 1.02664521779 rad
∠ C' = γ' = 93.58333216985° = 93°35' = 1.5088255565 rad

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How did we calculate this triangle?

1. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 12+18+21 = 51 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 51 }{ 2 } = 25.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 25.5 * (25.5-12)(25.5-18)(25.5-21) } ; ; T = sqrt{ 11618.44 } = 107.79 ; ;

4. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 107.79 }{ 12 } = 17.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 107.79 }{ 18 } = 11.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 107.79 }{ 21 } = 10.27 ; ;

5. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; alpha = arccos( fraction{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 18**2+21**2-12**2 }{ 2 * 18 * 21 } ) = 34° 46'19" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 12**2+21**2-18**2 }{ 2 * 12 * 21 } ) = 58° 48'41" ; ;
 gamma = 180° - alpha - beta = 180° - 34° 46'19" - 58° 48'41" = 86° 25' ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 107.79 }{ 25.5 } = 4.23 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 12 }{ 2 * sin 34° 46'19" } = 10.52 ; ;

8. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 18**2+2 * 21**2 - 12**2 } }{ 2 } = 18.615 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 21**2+2 * 12**2 - 18**2 } }{ 2 } = 14.543 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 18**2+2 * 12**2 - 21**2 } }{ 2 } = 11.124 ; ;
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