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 and angle β.

Right scalene triangle.

Sides: a = 39.12659040294   b = 8.31664676327   c = 40

Area: T = 162.695465723
Perimeter: p = 87.44223716621
Semiperimeter: s = 43.7211185831

Angle ∠ A = α = 78° = 1.36113568166 rad
Angle ∠ B = β = 12° = 0.20994395102 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 8.31664676327
Height: hb = 39.12659040294
Height: hc = 8.13547328615

Median: ma = 21.25772981683
Median: mb = 39.34662485453
Median: mc = 20

Inradius: r = 3.7211185831
Circumradius: R = 20

Vertex coordinates: A[40; 0] B[0; 0] C[38.27109091529; 8.13547328615]
Centroid: CG[26.0990303051; 2.71215776205]
Coordinates of the circumscribed circle: U[20; 0]
Coordinates of the inscribed circle: I[35.40547181983; 3.7211185831]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 102° = 1.36113568166 rad
∠ B' = β' = 168° = 0.20994395102 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

c = 40 ; ; beta = 12° ; ;

2. From angle β we calculate angle α:

 alpha + beta + 90° = 180° ; ; alpha = 90° - beta = 90° - 12 ° = 78 ° ; ;

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

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(78 ° ) = 39.126 ; ;

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{ 40**2 - 39.126**2 } = 8.316 ; ;


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 = 39.13 ; ; b = 8.32 ; ; c = 40 ; ;

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

p = a+b+c = 39.13+8.32+40 = 87.44 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 87.44 }{ 2 } = 43.72 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 43.72 * (43.72-39.13)(43.72-8.32)(43.72-40) } ; ; T = sqrt{ 26469.55 } = 162.69 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 162.69 }{ 39.13 } = 8.32 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 162.69 }{ 8.32 } = 39.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 162.69 }{ 40 } = 8.13 ; ;

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{ 39.13**2-8.32**2-40**2 }{ 2 * 8.32 * 40 } ) = 78° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 8.32**2-39.13**2-40**2 }{ 2 * 39.13 * 40 } ) = 12° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 40**2-39.13**2-8.32**2 }{ 2 * 8.32 * 39.13 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 162.69 }{ 43.72 } = 3.72 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 39.13 }{ 2 * sin 78° } = 20 ; ;
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