Triangle calculator SAS

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


Acute isosceles triangle.

Sides: a = 120   b = 120   c = 91.84440237676

Area: T = 5091.169882454
Perimeter: p = 331.8444023768
Semiperimeter: s = 165.9222011884

Angle ∠ A = α = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ B = β = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 84.85328137424
Height: hb = 84.85328137424
Height: hc = 110.8665543901

Median: ma = 88.41875454925
Median: mb = 88.41875454925
Median: mc = 110.8665543901

Inradius: r = 30.6844107351
Circumradius: R = 64.94435320175

Vertex coordinates: A[91.84440237676; 0] B[0; 0] C[45.92220118838; 110.8665543901]
Centroid: CG[45.92220118838; 36.95551813005]
Coordinates of the circumscribed circle: U[45.92220118838; 45.92220118838]
Coordinates of the inscribed circle: I[45.92220118838; 30.6844107351]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.5° = 112°30' = 1.17880972451 rad
∠ B' = β' = 112.5° = 112°30' = 1.17880972451 rad
∠ C' = γ' = 135° = 0.78553981634 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 = 120 ; ; b = 120 ; ; gamma = 45° ; ; ; ; c**2 = a**2+b**2 - 2ab cos( gamma ) ; ; c = sqrt{ a**2+b**2 - 2ab cos( gamma ) } ; ; c = sqrt{ 120**2+120**2 - 2 * 120 * 120 * cos(45° ) } ; ; c = 91.84 ; ;


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 = 120 ; ; b = 120 ; ; c = 91.84 ; ;

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

p = a+b+c = 120+120+91.84 = 331.84 ; ;

3. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 331.84 }{ 2 } = 165.92 ; ;

4. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 165.92 * (165.92-120)(165.92-120)(165.92-91.84) } ; ; T = sqrt{ 25920000 } = 5091.17 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5091.17 }{ 120 } = 84.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5091.17 }{ 120 } = 84.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5091.17 }{ 91.84 } = 110.87 ; ;

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{ 120**2-120**2-91.84**2 }{ 2 * 120 * 91.84 } ) = 67° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 120**2-120**2-91.84**2 }{ 2 * 120 * 91.84 } ) = 67° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 91.84**2-120**2-120**2 }{ 2 * 120 * 120 } ) = 45° ; ;

7. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5091.17 }{ 165.92 } = 30.68 ; ;

8. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 120 }{ 2 * sin 67° 30' } = 64.94 ; ;




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