Triangle calculator SAS

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


Acute scalene triangle.

Sides: a = 50   b = 65   c = 46.13989848424

Area: T = 1149.049851943
Perimeter: p = 161.1398984842
Semiperimeter: s = 80.56994924212

Angle ∠ A = α = 50.02109196315° = 50°1'15″ = 0.87330297424 rad
Angle ∠ B = β = 84.97990803685° = 84°58'45″ = 1.48331647477 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 45.96219407771
Height: hb = 35.35553390593
Height: hc = 49.80881404848

Median: ma = 50.51663633009
Median: mb = 35.47704519445
Median: mc = 53.20105499918

Inradius: r = 14.26215831985
Circumradius: R = 32.62551890591

Vertex coordinates: A[46.13989848424; 0] B[0; 0] C[4.37659732; 49.80881404848]
Centroid: CG[16.83883193475; 16.60327134949]
Coordinates of the circumscribed circle: U[23.06994924212; 23.06994924212]
Coordinates of the inscribed circle: I[15.56994924212; 14.26215831985]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.9799080369° = 129°58'45″ = 0.87330297424 rad
∠ B' = β' = 95.02109196315° = 95°1'15″ = 1.48331647477 rad
∠ C' = γ' = 135° = 0.78553981634 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 = 50 ; ; b = 65 ; ; gamma = 45° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 50**2+65**2 - 2 * 50 * 65 * cos(45° ) } ; ; c = 46.14 ; ;


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 = 50 ; ; b = 65 ; ; c = 46.14 ; ;

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

p = a+b+c = 50+65+46.14 = 161.14 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 161.14 }{ 2 } = 80.57 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 80.57 * (80.57-50)(80.57-65)(80.57-46.14) } ; ; T = sqrt{ 1320312.5 } = 1149.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1149.05 }{ 50 } = 45.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1149.05 }{ 65 } = 35.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1149.05 }{ 46.14 } = 49.81 ; ;

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{ 50**2-65**2-46.14**2 }{ 2 * 65 * 46.14 } ) = 50° 1'15" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 65**2-50**2-46.14**2 }{ 2 * 50 * 46.14 } ) = 84° 58'45" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 46.14**2-50**2-65**2 }{ 2 * 65 * 50 } ) = 45° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1149.05 }{ 80.57 } = 14.26 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 50 }{ 2 * sin 50° 1'15" } = 32.63 ; ;




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