Triangle calculator

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

You have entered side a, angle α and angle β.

Right scalene triangle.

Sides: a = 83   b = 101.3244290867   c = 58.11772256714

Area: T = 2411.865486536
Perimeter: p = 242.4421516539
Semiperimeter: s = 121.2210758269

Angle ∠ A = α = 55° = 0.96599310886 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 35° = 0.61108652382 rad

Height: ha = 58.11772256714
Height: hb = 47.60768442171
Height: hc = 83

Median: ma = 71.41333175237
Median: mb = 50.66221454336
Median: mc = 87.94397690464

Inradius: r = 19.89664674021
Circumradius: R = 50.66221454336

Vertex coordinates: A[58.11772256714; 0] B[0; 0] C[-0; 83]
Centroid: CG[19.37224085571; 27.66766666667]
Coordinates of the circumscribed circle: U[29.05986128357; 41.5]
Coordinates of the inscribed circle: I[19.89664674021; 19.89664674021]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125° = 0.96599310886 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 145° = 0.61108652382 rad

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How did we calculate this triangle?

1. Input data entered: side a, angle α and angle β.

a = 83 ; ; alpha = 55° ; ; beta = 90° ; ;

2. From angle α and angle β we calculate angle γ:

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 55 ° - 90 ° = 35 ° ; ;

3. From angle β, angle α and side a we calculate side b - By using the law of sines, we calculate unknown side b:

 fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 83 * fraction{ sin 90° }{ sin 55° } = 101.32 ; ;

4. From angle γ, angle α and side a we calculate side c - By using the law of sines, we calculate unknown side c:

 fraction{ c }{ a } = fraction{ sin gamma }{ sin alpha } ; ; ; ; c = a * fraction{ sin gamma }{ sin alpha } ; ; ; ; c = 83 * fraction{ sin 35° }{ sin 55° } = 58.12 ; ;
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 = 83 ; ; b = 101.32 ; ; c = 58.12 ; ;

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

p = a+b+c = 83+101.32+58.12 = 242.44 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 242.44 }{ 2 } = 121.22 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 121.22 * (121.22-83)(121.22-101.32)(121.22-58.12) } ; ; T = sqrt{ 5817092.13 } = 2411.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2411.86 }{ 83 } = 58.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2411.86 }{ 101.32 } = 47.61 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2411.86 }{ 58.12 } = 83 ; ;

9. 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{ 101.32**2+58.12**2-83**2 }{ 2 * 101.32 * 58.12 } ) = 55° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 83**2+58.12**2-101.32**2 }{ 2 * 83 * 58.12 } ) = 90° ; ; gamma = 180° - alpha - beta = 180° - 55° - 90° = 35° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2411.86 }{ 121.22 } = 19.9 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 83 }{ 2 * sin 55° } = 50.66 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 101.32**2+2 * 58.12**2 - 83**2 } }{ 2 } = 71.413 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 58.12**2+2 * 83**2 - 101.32**2 } }{ 2 } = 50.662 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 101.32**2+2 * 83**2 - 58.12**2 } }{ 2 } = 87.94 ; ;
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