Triangle calculator SAS

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


Obtuse isosceles triangle.

Sides: a = 8.35   b = 8.35   c = 12.79329422001

Area: T = 34.33216292797
Perimeter: p = 29.49329422001
Semiperimeter: s = 14.74664711

Angle ∠ A = α = 40° = 0.69881317008 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 100° = 1.7455329252 rad

Height: ha = 8.22331447377
Height: hb = 8.22331447377
Height: hc = 5.36772765409

Median: ma = 9.96329468566
Median: mb = 9.96329468566
Median: mc = 5.36772765409

Inradius: r = 2.32881250848
Circumradius: R = 6.49551469771

Vertex coordinates: A[12.79329422001; 0] B[0; 0] C[6.39664711; 5.36772765409]
Centroid: CG[6.39664711; 1.78990921803]
Coordinates of the circumscribed circle: U[6.39664711; -1.12878704363]
Coordinates of the inscribed circle: I[6.39664711; 2.32881250848]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140° = 0.69881317008 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 80° = 1.7455329252 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 = 8.35 ; ; b = 8.35 ; ; gamma = 100° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 8.35**2+8.35**2 - 2 * 8.35 * 8.35 * cos(100° ) } ; ; c = 12.79 ; ;


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 = 8.35 ; ; b = 8.35 ; ; c = 12.79 ; ;

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

p = a+b+c = 8.35+8.35+12.79 = 29.49 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 29.49 }{ 2 } = 14.75 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 14.75 * (14.75-8.35)(14.75-8.35)(14.75-12.79) } ; ; T = sqrt{ 1178.66 } = 34.33 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 34.33 }{ 8.35 } = 8.22 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 34.33 }{ 8.35 } = 8.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 34.33 }{ 12.79 } = 5.37 ; ;

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{ 8.35**2-8.35**2-12.79**2 }{ 2 * 8.35 * 12.79 } ) = 40° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.35**2-8.35**2-12.79**2 }{ 2 * 8.35 * 12.79 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 12.79**2-8.35**2-8.35**2 }{ 2 * 8.35 * 8.35 } ) = 100° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 34.33 }{ 14.75 } = 2.33 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.35 }{ 2 * sin 40° } = 6.5 ; ;




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