Right triangle calculator (B,b)

Please enter two properties of the right triangle

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


You have entered cathetus b and angle β.

Right scalene triangle.

Sides: a = 375.6222486979   b = 517   c = 639.0477144367

Area: T = 97098.4132884
Perimeter: p = 1531.676963135
Semiperimeter: s = 765.8354815673

Angle ∠ A = α = 36° = 0.62883185307 rad
Angle ∠ B = β = 54° = 0.94224777961 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 517
Height: hb = 375.6222486979
Height: hc = 303.8854975435

Median: ma = 550.056641818
Median: mb = 455.9766427816
Median: mc = 319.5243572184

Inradius: r = 126.7887671306
Circumradius: R = 319.5243572184

Vertex coordinates: A[639.0477144367; 0] B[0; 0] C[220.7855358275; 303.8854975435]
Centroid: CG[286.6110834214; 101.2954991812]
Coordinates of the circumscribed circle: U[319.5243572184; -0]
Coordinates of the inscribed circle: I[248.8354815673; 126.7887671306]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 144° = 0.62883185307 rad
∠ B' = β' = 126° = 0.94224777961 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus b angle β

b = 517 ; ; beta = 54° ; ;

2. From angle β we calculate angle α:

 alpha + beta + 90° = 180° ; ; alpha = 90° - beta = 90° - 54 ° = 36 ° ; ;

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

 cos alpha = b:c ; ; c = b/ cos alpha = b/ cos(36 ° ) = 639.047 ; ;

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

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(36 ° ) = 375.622 ; ;


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 = 375.62 ; ; b = 517 ; ; c = 639.05 ; ;

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

p = a+b+c = 375.62+517+639.05 = 1531.67 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1531.67 }{ 2 } = 765.83 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 765.83 * (765.83-375.62)(765.83-517)(765.83-639.05) } ; ; T = sqrt{ 9428101784.59 } = 97098.41 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 97098.41 }{ 375.62 } = 517 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 97098.41 }{ 517 } = 375.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 97098.41 }{ 639.05 } = 303.88 ; ;

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{ 375.62**2-517**2-639.05**2 }{ 2 * 517 * 639.05 } ) = 36° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 517**2-375.62**2-639.05**2 }{ 2 * 375.62 * 639.05 } ) = 54° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 639.05**2-375.62**2-517**2 }{ 2 * 517 * 375.62 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 97098.41 }{ 765.83 } = 126.79 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 375.62 }{ 2 * sin 36° } = 319.52 ; ;
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