Triangle calculator

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 = 1083.462158215   b = 417   c = 1000

Area: T = 208500
Perimeter: p = 2500.462158215
Semiperimeter: s = 1250.231079108

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 22.63661363795° = 22°38'10″ = 0.39550751098 rad
Angle ∠ C = γ = 67.36438636205° = 67°21'50″ = 1.1765721217 rad

Height: ha = 384.8787513766
Height: hb = 1000
Height: hc = 417

Median: ma = 541.7310791076
Median: mb = 1021.505489475
Median: mc = 651.0687584817

Inradius: r = 166.7699208924
Circumradius: R = 541.7310791076

Vertex coordinates: A[1000; 0] B[0; 0] C[1000; 417]
Centroid: CG[666.6676666667; 139]
Coordinates of the circumscribed circle: U[500; 208.5]
Coordinates of the inscribed circle: I[833.2310791076; 166.7699208924]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 157.3643863621° = 157°21'50″ = 0.39550751098 rad
∠ C' = γ' = 112.6366136379° = 112°38'10″ = 1.1765721217 rad

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How did we calculate this triangle?

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

b = 417 ; ; c = 1000 ; ; 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{ 417**2+1000**2 - 2 * 417 * 1000 * cos 90° } ; ; a = 1083.46 ; ;


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 = 1083.46 ; ; b = 417 ; ; c = 1000 ; ;

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

p = a+b+c = 1083.46+417+1000 = 2500.46 ; ;

4. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2500.46 }{ 2 } = 1250.23 ; ;

5. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1250.23 * (1250.23-1083.46)(1250.23-417)(1250.23-1000) } ; ; T = sqrt{ 43472250000 } = 208500 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 208500 }{ 1083.46 } = 384.88 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 208500 }{ 417 } = 1000 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 208500 }{ 1000 } = 417 ; ;

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{ 417**2+1000**2-1083.46**2 }{ 2 * 417 * 1000 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 1083.46**2+1000**2-417**2 }{ 2 * 1083.46 * 1000 } ) = 22° 38'10" ; ; gamma = 180° - alpha - beta = 180° - 90° - 22° 38'10" = 67° 21'50" ; ;

8. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 208500 }{ 1250.23 } = 166.77 ; ;

9. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 1083.46 }{ 2 * sin 90° } = 541.73 ; ;

10. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 417**2+2 * 1000**2 - 1083.46**2 } }{ 2 } = 541.731 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 1000**2+2 * 1083.46**2 - 417**2 } }{ 2 } = 1021.505 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 417**2+2 * 1083.46**2 - 1000**2 } }{ 2 } = 651.068 ; ;
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