Triangle calculator SAS

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


Obtuse isosceles triangle.

Sides: a = 40   b = 40   c = 69.28220323028

Area: T = 692.8220323028
Perimeter: p = 149.2822032303
Semiperimeter: s = 74.64110161514

Angle ∠ A = α = 30° = 0.52435987756 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: ha = 34.64110161514
Height: hb = 34.64110161514
Height: hc = 20

Median: ma = 52.91550262213
Median: mb = 52.91550262213
Median: mc = 20

Inradius: r = 9.28220323028
Circumradius: R = 40

Vertex coordinates: A[69.28220323028; 0] B[0; 0] C[34.64110161514; 20]
Centroid: CG[34.64110161514; 6.66766666667]
Coordinates of the circumscribed circle: U[34.64110161514; -20]
Coordinates of the inscribed circle: I[34.64110161514; 9.28220323028]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150° = 0.52435987756 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 60° = 2.09443951024 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 = 40 ; ; b = 40 ; ; gamma = 120° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 40**2+40**2 - 2 * 40 * 40 * cos(120° ) } ; ; c = 69.28 ; ;


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 = 40 ; ; b = 40 ; ; c = 69.28 ; ;

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

p = a+b+c = 40+40+69.28 = 149.28 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 149.28 }{ 2 } = 74.64 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 74.64 * (74.64-40)(74.64-40)(74.64-69.28) } ; ; T = sqrt{ 480000 } = 692.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 692.82 }{ 40 } = 34.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 692.82 }{ 40 } = 34.64 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 692.82 }{ 69.28 } = 20 ; ;

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{ 40**2-40**2-69.28**2 }{ 2 * 40 * 69.28 } ) = 30° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 40**2-40**2-69.28**2 }{ 2 * 40 * 69.28 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 69.28**2-40**2-40**2 }{ 2 * 40 * 40 } ) = 120° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 692.82 }{ 74.64 } = 9.28 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 40 }{ 2 * sin 30° } = 40 ; ;




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