Right triangle calculator (a,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 = 0.30771859917   b = 0.24   c = 0.39898246189

Area: T = 0.0376862319
Perimeter: p = 0.93770106106
Semiperimeter: s = 0.46985053053

Angle ∠ A = α = 52° = 0.9087571211 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 0.24
Height: hb = 0.30771859917
Height: hc = 0.18991225809

Median: ma = 0.28549400084
Median: mb = 0.33297927129
Median: mc = 0.19549123095

Inradius: r = 0.07986806864
Circumradius: R = 0.19549123095

Vertex coordinates: A[0.39898246189; 0] B[0; 0] C[0.24220658648; 0.18991225809]
Centroid: CG[0.21106301613; 0.06330408603]
Coordinates of the circumscribed circle: U[0.19549123095; -0]
Coordinates of the inscribed circle: I[0.22985053053; 0.07986806864]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 128° = 0.9087571211 rad
∠ B' = β' = 142° = 0.66332251158 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 = 0.24 ; ; beta = 38° ; ;

2. From angle β we calculate angle α:

 alpha + beta + 90° = 180° ; ; alpha = 90° - beta = 90° - 38 ° = 52 ° ; ;

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

 cos alpha = b:c ; ; c = b/ cos alpha = b/ cos(52 ° ) = 0.39 ; ;

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

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(52 ° ) = 0.307 ; ;


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 = 0.31 ; ; b = 0.24 ; ; c = 0.39 ; ;

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

p = a+b+c = 0.31+0.24+0.39 = 0.94 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 0.94 }{ 2 } = 0.47 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 0.47 * (0.47-0.31)(0.47-0.24)(0.47-0.39) } ; ; T = sqrt{ 0 } = 0.04 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.04 }{ 0.31 } = 0.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.04 }{ 0.24 } = 0.31 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.04 }{ 0.39 } = 0.19 ; ;

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{ 0.31**2-0.24**2-0.39**2 }{ 2 * 0.24 * 0.39 } ) = 52° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 0.24**2-0.31**2-0.39**2 }{ 2 * 0.31 * 0.39 } ) = 38° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 0.39**2-0.31**2-0.24**2 }{ 2 * 0.24 * 0.31 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.04 }{ 0.47 } = 0.08 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 0.31 }{ 2 * sin 52° } = 0.19 ; ;
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