Triangle calculator SAS

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


Right isosceles triangle.

Sides: a = 60   b = 60   c = 84.85328137424

Area: T = 1800
Perimeter: p = 204.8532813742
Semiperimeter: s = 102.4266406871

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 45° = 0.78553981634 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 60
Height: hb = 60
Height: hc = 42.42664068712

Median: ma = 67.0822039325
Median: mb = 67.0822039325
Median: mc = 42.42664068712

Inradius: r = 17.57435931288
Circumradius: R = 42.42664068712

Vertex coordinates: A[84.85328137424; 0] B[0; 0] C[42.42664068712; 42.42664068712]
Centroid: CG[42.42664068712; 14.14221356237]
Coordinates of the circumscribed circle: U[42.42664068712; -0]
Coordinates of the inscribed circle: I[42.42664068712; 17.57435931288]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 135° = 0.78553981634 rad
∠ C' = γ' = 90° = 1.57107963268 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 = 60 ; ; b = 60 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 60**2+60**2 - 2 * 60 * 60 * cos(90° ) } ; ; c = 84.85 ; ;


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 = 60 ; ; b = 60 ; ; c = 84.85 ; ;

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

p = a+b+c = 60+60+84.85 = 204.85 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 204.85 }{ 2 } = 102.43 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 102.43 * (102.43-60)(102.43-60)(102.43-84.85) } ; ; T = sqrt{ 3240000 } = 1800 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1800 }{ 60 } = 60 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1800 }{ 60 } = 60 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1800 }{ 84.85 } = 42.43 ; ;

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{ 60**2-60**2-84.85**2 }{ 2 * 60 * 84.85 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 60**2-60**2-84.85**2 }{ 2 * 60 * 84.85 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 84.85**2-60**2-60**2 }{ 2 * 60 * 60 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1800 }{ 102.43 } = 17.57 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 60 }{ 2 * sin 45° } = 42.43 ; ;




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