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 and 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 = 40 * 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 - from two legs

T = fraction{ ab }{ 2 } = fraction{ 39.13 * 8.32 }{ 2 } = 162.69 ; ;

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

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

9. Calculation of the inner angles of the triangle - basic use of sine function

 sin alpha = fraction{ a }{ c } ; ; alpha = arcsin( fraction{ a }{ c } ) = arcsin( fraction{ 39.13 }{ 40 } ) = 78° ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 8.32 }{ 40 } ) = 12° ; ; gamma = 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 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.32**2+2 * 40**2 - 39.13**2 } }{ 2 } = 21.257 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 40**2+2 * 39.13**2 - 8.32**2 } }{ 2 } = 39.346 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 8.32**2+2 * 39.13**2 - 40**2 } }{ 2 } = 20 ; ;
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Trigonometry right triangle solver. You can calculate angles, sides (adjacent, opposite, hypotenuse) and area of any right-angled triangle and use it in real world to find height. A right-angled triangle is entirely determined by two independent properties. Step-by-step explanations are provided for each calculation.

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