Triangle calculator SAS

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


Right isosceles triangle.

Sides: a = 93   b = 93   c = 131.5221861301

Area: T = 4324.5
Perimeter: p = 317.5221861301
Semiperimeter: s = 158.761093065

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

Height: ha = 93
Height: hb = 93
Height: hc = 65.76109306503

Median: ma = 103.9777160954
Median: mb = 103.9777160954
Median: mc = 65.76109306503

Inradius: r = 27.23990693497
Circumradius: R = 65.76109306503

Vertex coordinates: A[131.5221861301; 0] B[0; 0] C[65.76109306503; 65.76109306503]
Centroid: CG[65.76109306503; 21.92203102168]
Coordinates of the circumscribed circle: U[65.76109306503; -0]
Coordinates of the inscribed circle: I[65.76109306503; 27.23990693497]

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 = 93 ; ; b = 93 ; ; gamma = 90° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 93**2+93**2 - 2 * 93 * 93 * cos(90° ) } ; ; c = 131.52 ; ;


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 = 93 ; ; b = 93 ; ; c = 131.52 ; ;

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

p = a+b+c = 93+93+131.52 = 317.52 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 317.52 }{ 2 } = 158.76 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 158.76 * (158.76-93)(158.76-93)(158.76-131.52) } ; ; T = sqrt{ 18701300.25 } = 4324.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4324.5 }{ 93 } = 93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4324.5 }{ 93 } = 93 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4324.5 }{ 131.52 } = 65.76 ; ;

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{ 93**2-93**2-131.52**2 }{ 2 * 93 * 131.52 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 93**2-93**2-131.52**2 }{ 2 * 93 * 131.52 } ) = 45° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 131.52**2-93**2-93**2 }{ 2 * 93 * 93 } ) = 90° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4324.5 }{ 158.76 } = 27.24 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 93 }{ 2 * sin 45° } = 65.76 ; ;




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