Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 90   b = 100   c = 80.80773203716

Area: T = 3447.219999404
Perimeter: p = 270.8077320372
Semiperimeter: s = 135.4043660186

Angle ∠ A = α = 58.56603483671° = 58°33'37″ = 1.02220708901 rad
Angle ∠ B = β = 71.44396516329° = 71°26'23″ = 1.24768571375 rad
Angle ∠ C = γ = 50° = 0.8732664626 rad

Height: ha = 76.60444443119
Height: hb = 68.94439998807
Height: hc = 85.31990027384

Median: ma = 78.99331105402
Median: mb = 69.39895634287
Median: mc = 86.12551661455

Inradius: r = 25.45986913626
Circumradius: R = 52.74332325223

Vertex coordinates: A[80.80773203716; 0] B[0; 0] C[28.64772995537; 85.31990027384]
Centroid: CG[36.48548733085; 28.44396675795]
Coordinates of the circumscribed circle: U[40.40436601858; 33.90326963601]
Coordinates of the inscribed circle: I[35.40436601858; 25.45986913626]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 121.4439651633° = 121°26'23″ = 1.02220708901 rad
∠ B' = β' = 108.5660348367° = 108°33'37″ = 1.24768571375 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 = 100 ; ; gamma = 50° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 90**2+100**2 - 2 * 90 * 100 * cos(50° ) } ; ; c = 80.81 ; ;


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 = 100 ; ; c = 80.81 ; ;

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

p = a+b+c = 90+100+80.81 = 270.81 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 270.81 }{ 2 } = 135.4 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 135.4 * (135.4-90)(135.4-100)(135.4-80.81) } ; ; T = sqrt{ 11883187.8 } = 3447.2 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3447.2 }{ 90 } = 76.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3447.2 }{ 100 } = 68.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3447.2 }{ 80.81 } = 85.32 ; ;

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{ 90**2-100**2-80.81**2 }{ 2 * 100 * 80.81 } ) = 58° 33'37" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 100**2-90**2-80.81**2 }{ 2 * 90 * 80.81 } ) = 71° 26'23" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 80.81**2-90**2-100**2 }{ 2 * 100 * 90 } ) = 50° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3447.2 }{ 135.4 } = 25.46 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 90 }{ 2 * sin 58° 33'37" } = 52.74 ; ;




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