Triangle calculator SAS

Please enter two sides of the triangle and the included angle
°


Acute scalene triangle.

Sides: a = 90   b = 80   c = 72.41444904043

Area: T = 2757.765999523
Perimeter: p = 242.4144490404
Semiperimeter: s = 121.2077245202

Angle ∠ A = α = 72.19897406737° = 72°11'23″ = 1.26599486609 rad
Angle ∠ B = β = 57.81102593263° = 57°48'37″ = 1.00989793667 rad
Angle ∠ C = γ = 50° = 0.8732664626 rad

Height: ha = 61.28435554495
Height: hb = 68.94439998807
Height: hc = 76.16659711981

Median: ma = 61.61992276019
Median: mb = 71.21774782638
Median: mc = 77.06551373506

Inradius: r = 22.75224352247
Circumradius: R = 47.26552018135

Vertex coordinates: A[72.41444904043; 0] B[0; 0] C[47.94552274106; 76.16659711981]
Centroid: CG[40.12199059383; 25.3898657066]
Coordinates of the circumscribed circle: U[36.20772452021; 30.38114860951]
Coordinates of the inscribed circle: I[41.20772452021; 22.75224352247]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 107.8110259326° = 107°48'37″ = 1.26599486609 rad
∠ B' = β' = 122.1989740674° = 122°11'23″ = 1.00989793667 rad
∠ C' = γ' = 130° = 0.8732664626 rad

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How did we calculate this triangle?

1. Calculation of the third side c of the triangle using a Law of Cosines

a = 90 ; ; b = 80 ; ; gamma = 50° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 90**2+80**2 - 2 * 90 * 80 * cos 50° } ; ; c = 72.41 ; ;
Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 90 ; ; b = 80 ; ; c = 72.41 ; ;

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

p = a+b+c = 90+80+72.41 = 242.41 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 242.41 }{ 2 } = 121.21 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 121.21 * (121.21-90)(121.21-80)(121.21-72.41) } ; ; T = sqrt{ 7605240.19 } = 2757.76 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2757.76 }{ 90 } = 61.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2757.76 }{ 80 } = 68.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2757.76 }{ 72.41 } = 76.17 ; ;

6. 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{ 80**2+72.41**2-90**2 }{ 2 * 80 * 72.41 } ) = 72° 11'23" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 90**2+72.41**2-80**2 }{ 2 * 90 * 72.41 } ) = 57° 48'37" ; ;
 gamma = 180° - alpha - beta = 180° - 72° 11'23" - 57° 48'37" = 50° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2757.76 }{ 121.21 } = 22.75 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 90 }{ 2 * sin 72° 11'23" } = 47.27 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 72.41**2 - 90**2 } }{ 2 } = 61.619 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 72.41**2+2 * 90**2 - 80**2 } }{ 2 } = 71.217 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 80**2+2 * 90**2 - 72.41**2 } }{ 2 } = 77.065 ; ;
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