Triangle calculator - result

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side b, c and angle α.

Right scalene triangle.

Sides: a = 68.96437585983   b = 34   c = 60

Area: T = 1020
Perimeter: p = 162.9643758598
Semiperimeter: s = 81.48218792991

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 29.53987822596° = 29°32'20″ = 0.51655490075 rad
Angle ∠ C = γ = 60.46112177404° = 60°27'40″ = 1.05552473193 rad

Height: ha = 29.58107543189
Height: hb = 60
Height: hc = 34

Median: ma = 34.48218792991
Median: mb = 62.36218473107
Median: mc = 45.3433136195

Inradius: r = 12.51881207009
Circumradius: R = 34.48218792991

Vertex coordinates: A[60; 0] B[0; 0] C[60; 34]
Centroid: CG[40; 11.33333333333]
Coordinates of the circumscribed circle: U[30; 17]
Coordinates of the inscribed circle: I[47.48218792991; 12.51881207009]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 150.461121774° = 150°27'40″ = 0.51655490075 rad
∠ C' = γ' = 119.539878226° = 119°32'20″ = 1.05552473193 rad

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How did we calculate this triangle?

1. Input data entered: side b, c and angle α.

b = 34 ; ; c = 60 ; ; alpha = 90° ; ;

2. Calculation of the third side a of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos alpha ; ; a = sqrt{ b**2+c**2 - 2bc cos alpha } ; ; a = sqrt{ 34**2+60**2 - 2 * 34 * 60 * cos 90° } ; ; a = 68.96 ; ;


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 = 68.96 ; ; b = 34 ; ; c = 60 ; ;

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

p = a+b+c = 68.96+34+60 = 162.96 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 162.96 }{ 2 } = 81.48 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 81.48 * (81.48-68.96)(81.48-34)(81.48-60) } ; ; T = sqrt{ 1040400 } = 1020 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1020 }{ 68.96 } = 29.58 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1020 }{ 34 } = 60 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1020 }{ 60 } = 34 ; ;

7. 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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 34**2+60**2-68.96**2 }{ 2 * 34 * 60 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 68.96**2+60**2-34**2 }{ 2 * 68.96 * 60 } ) = 29° 32'20" ; ; gamma = 180° - alpha - beta = 180° - 90° - 29° 32'20" = 60° 27'40" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1020 }{ 81.48 } = 12.52 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 68.96 }{ 2 * sin 90° } = 34.48 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 60**2 - 68.96**2 } }{ 2 } = 34.482 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 60**2+2 * 68.96**2 - 34**2 } }{ 2 } = 62.362 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 34**2+2 * 68.96**2 - 60**2 } }{ 2 } = 45.343 ; ;
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