Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 73   b = 56   c = 121.3659710641

Area: T = 1313.85878742
Perimeter: p = 250.3659710641
Semiperimeter: s = 125.1879855321

Angle ∠ A = α = 22.74660914146° = 22°44'46″ = 0.39769941871 rad
Angle ∠ B = β = 17.25439085854° = 17°15'14″ = 0.30111375137 rad
Angle ∠ C = γ = 140° = 2.44334609528 rad

Height: ha = 35.99661061424
Height: hb = 46.92334955071
Height: hc = 21.6522290818

Median: ma = 87.17770593876
Median: mb = 96.1498789298
Median: mc = 23.46217807991

Inradius: r = 10.49657612456
Circumradius: R = 94.40110967328

Vertex coordinates: A[121.3659710641; 0] B[0; 0] C[69.71549790384; 21.6522290818]
Centroid: CG[63.69215632266; 7.21774302727]
Coordinates of the circumscribed circle: U[60.68798553207; -72.31554355765]
Coordinates of the inscribed circle: I[69.18798553207; 10.49657612456]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 157.2543908585° = 157°15'14″ = 0.39769941871 rad
∠ B' = β' = 162.7466091415° = 162°44'46″ = 0.30111375137 rad
∠ C' = γ' = 40° = 2.44334609528 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 = 73 ; ; b = 56 ; ; gamma = 140° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 73**2+56**2 - 2 * 73 * 56 * cos(140° ) } ; ; c = 121.36 ; ;


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 = 73 ; ; b = 56 ; ; c = 121.36 ; ;

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

p = a+b+c = 73+56+121.36 = 250.36 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 250.36 }{ 2 } = 125.18 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 125.18 * (125.18-73)(125.18-56)(125.18-121.36) } ; ; T = sqrt{ 1726222.51 } = 1313.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1313.86 }{ 73 } = 36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1313.86 }{ 56 } = 46.92 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1313.86 }{ 121.36 } = 21.65 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 73**2-56**2-121.36**2 }{ 2 * 56 * 121.36 } ) = 22° 44'46" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 56**2-73**2-121.36**2 }{ 2 * 73 * 121.36 } ) = 17° 15'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 121.36**2-73**2-56**2 }{ 2 * 56 * 73 } ) = 140° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1313.86 }{ 125.18 } = 10.5 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 73 }{ 2 * sin 22° 44'46" } = 94.4 ; ;




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