Triangle calculator SAS

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


Acute isosceles triangle.

Sides: a = 90   b = 90   c = 90

Area: T = 3507.403288533
Perimeter: p = 270
Semiperimeter: s = 135

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 60° = 1.04771975512 rad
Angle ∠ C = γ = 60° = 1.04771975512 rad

Height: ha = 77.94222863406
Height: hb = 77.94222863406
Height: hc = 77.94222863406

Median: ma = 77.94222863406
Median: mb = 77.94222863406
Median: mc = 77.94222863406

Inradius: r = 25.98107621135
Circumradius: R = 51.96215242271

Vertex coordinates: A[90; 0] B[0; 0] C[45; 77.94222863406]
Centroid: CG[45; 25.98107621135]
Coordinates of the circumscribed circle: U[45; 25.98107621135]
Coordinates of the inscribed circle: I[45; 25.98107621135]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 120° = 1.04771975512 rad
∠ C' = γ' = 120° = 1.04771975512 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 = 90 ; ; gamma = 60° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 90**2+90**2 - 2 * 90 * 90 * cos(60° ) } ; ; c = 90 ; ;


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 = 90 ; ; c = 90 ; ;

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

p = a+b+c = 90+90+90 = 270 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 270 }{ 2 } = 135 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 135 * (135-90)(135-90)(135-90) } ; ; T = sqrt{ 12301875 } = 3507.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3507.4 }{ 90 } = 77.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3507.4 }{ 90 } = 77.94 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3507.4 }{ 90 } = 77.94 ; ;

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

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3507.4 }{ 135 } = 25.98 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 90 }{ 2 * sin 60° } = 51.96 ; ;




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