Right triangle calculator (B,c)

Please enter two properties of the right triangle

Use symbols: a, b, c, A, B, h, T, p, r, R


You have entered hypotenuse c, angle β and angle γ.

Right scalene triangle.

Sides: a = 83.13884387633   b = 48   c = 96

Area: T = 1995.323253032
Perimeter: p = 227.1388438763
Semiperimeter: s = 113.5699219382

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 30° = 0.52435987756 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 48
Height: hb = 83.13884387633
Height: hc = 41.56992193817

Median: ma = 63.49880314656
Median: mb = 86.53332306111
Median: mc = 48

Inradius: r = 17.56992193817
Circumradius: R = 48

Vertex coordinates: A[96; 0] B[0; 0] C[72; 41.56992193817]
Centroid: CG[56; 13.85664064606]
Coordinates of the circumscribed circle: U[48; 0]
Coordinates of the inscribed circle: I[65.56992193817; 17.56992193817]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 150° = 0.52435987756 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: hypotenuse c angle β angle γ

c = 96 ; ; beta = 30° ; ; gamma = 90° ; ;

2. From angle β we calculate angle α:

 alpha + beta + 90° = 180° ; ; alpha = 90° - beta = 90° - 30 ° = 60 ° ; ;

3. From hypotenuse c and angle α we calculate cathetus a:

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(60 ° ) = 83.138 ; ;

4. From cathetus a and hypotenuse c we calculate cathetus b - Pythagorean theorem:

c**2 = a**2+b**2 ; ; b = sqrt{ c**2 - a**2 } = sqrt{ 96**2 - 83.138**2 } = 48 ; ;


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 = 83.14 ; ; b = 48 ; ; c = 96 ; ;

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

p = a+b+c = 83.14+48+96 = 227.14 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 227.14 }{ 2 } = 113.57 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 113.57 * (113.57-83.14)(113.57-48)(113.57-96) } ; ; T = sqrt{ 3981312 } = 1995.32 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1995.32 }{ 83.14 } = 48 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1995.32 }{ 48 } = 83.14 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1995.32 }{ 96 } = 41.57 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 83.14**2-48**2-96**2 }{ 2 * 48 * 96 } ) = 60° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 48**2-83.14**2-96**2 }{ 2 * 83.14 * 96 } ) = 30° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 96**2-83.14**2-48**2 }{ 2 * 48 * 83.14 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1995.32 }{ 113.57 } = 17.57 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 83.14 }{ 2 * sin 60° } = 48 ; ;
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