Triangle calculator SAS

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


Obtuse scalene triangle.

Sides: a = 53   b = 93   c = 107.8432690168

Area: T = 2464.125464471
Perimeter: p = 253.8432690168
Semiperimeter: s = 126.9211345084

Angle ∠ A = α = 29.43114393103° = 29°25'53″ = 0.51436755196 rad
Angle ∠ B = β = 59.56985606897° = 59°34'7″ = 1.04396675147 rad
Angle ∠ C = γ = 91° = 1.58882496193 rad

Height: ha = 92.98658356495
Height: hb = 52.99219278433
Height: hc = 45.69985010458

Median: ma = 97.14656273402
Median: mb = 71.11545056323
Median: mc = 53.11876857961

Inradius: r = 19.41545802905
Circumradius: R = 53.9329558817

Vertex coordinates: A[107.8432690168; 0] B[0; 0] C[26.84548691962; 45.69985010458]
Centroid: CG[44.89658531214; 15.23328336819]
Coordinates of the circumscribed circle: U[53.9211345084; -0.94112005795]
Coordinates of the inscribed circle: I[33.9211345084; 19.41545802905]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150.569856069° = 150°34'7″ = 0.51436755196 rad
∠ B' = β' = 120.431143931° = 120°25'53″ = 1.04396675147 rad
∠ C' = γ' = 89° = 1.58882496193 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 = 53 ; ; b = 93 ; ; gamma = 91° ; ; ; ; c**2 = a**2+b**2 - 2ab cos gamma ; ; c = sqrt{ a**2+b**2 - 2ab cos gamma } ; ; c = sqrt{ 53**2+93**2 - 2 * 53 * 93 * cos 91° } ; ; c = 107.84 ; ;
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 = 53 ; ; b = 93 ; ; c = 107.84 ; ;

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

p = a+b+c = 53+93+107.84 = 253.84 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 253.84 }{ 2 } = 126.92 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 126.92 * (126.92-53)(126.92-93)(126.92-107.84) } ; ; T = sqrt{ 6071910.26 } = 2464.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2464.12 }{ 53 } = 92.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2464.12 }{ 93 } = 52.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2464.12 }{ 107.84 } = 45.7 ; ;

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{ 93**2+107.84**2-53**2 }{ 2 * 93 * 107.84 } ) = 29° 25'53" ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 53**2+107.84**2-93**2 }{ 2 * 53 * 107.84 } ) = 59° 34'7" ; ; gamma = 180° - alpha - beta = 180° - 29° 25'53" - 59° 34'7" = 91° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2464.12 }{ 126.92 } = 19.41 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 53 }{ 2 * sin 29° 25'53" } = 53.93 ; ;

9. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 93**2+2 * 107.84**2 - 53**2 } }{ 2 } = 97.146 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 107.84**2+2 * 53**2 - 93**2 } }{ 2 } = 71.115 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 93**2+2 * 53**2 - 107.84**2 } }{ 2 } = 53.118 ; ;
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