Right triangle calculator (B,h)

Please enter two properties of the right triangle

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


You have entered height h and angle β.

Right scalene triangle.

Sides: a = 13.06656296488   b = 5.41219610015   c = 14.14221356237

Area: T = 35.35553390593
Perimeter: p = 32.6219726274
Semiperimeter: s = 16.3109863137

Angle ∠ A = α = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ B = β = 22.5° = 22°30' = 0.39326990817 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 5.41219610015
Height: hb = 13.06656296488
Height: hc = 5

Median: ma = 8.48333361015
Median: mb = 13.3432901056
Median: mc = 7.07110678119

Inradius: r = 2.16877275132
Circumradius: R = 7.07110678119

Vertex coordinates: A[14.14221356237; 0] B[0; 0] C[12.07110678119; 5]
Centroid: CG[8.73877344785; 1.66766666667]
Coordinates of the circumscribed circle: U[7.07110678119; -0]
Coordinates of the inscribed circle: I[10.89879021355; 2.16877275132]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 112.5° = 112°30' = 1.17880972451 rad
∠ B' = β' = 157.5° = 157°30' = 0.39326990817 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: angle β and height h

 beta = 22.5° ; ; h = 5 ; ;

2. From angle β we calculate angle α:

 alpha + beta + 90° = 180° ; ; alpha = 90° - beta = 90° - 22.5 ° = 67.5 ° ; ;

3. From height h and angle α we calculate cathetus a:

a = fraction{ h }{ sin beta } = fraction{ 5 }{ sin (22° 30') } = 13.066 ; ; b = fraction{ h }{ sin alpha } = fraction{ 5 }{ sin (67° 30') } = 5.412 ; ;

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

 sin alpha = a:c ; ; c = a/ sin alpha = 13.066/ sin(67.5 ° ) = 14.142 ; ;


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 = 13.07 ; ; b = 5.41 ; ; c = 14.14 ; ;

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

p = a+b+c = 13.07+5.41+14.14 = 32.62 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 32.62 }{ 2 } = 16.31 ; ;

7. The triangle area - from two legs

T = fraction{ ab }{ 2 } = fraction{ 13.07 * 5.41 }{ 2 } = 35.36 ; ;

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

h _a = b = 5.41 ; ; h _b = a = 13.07 ; ; T = fraction{ c h _c }{ 2 } ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 35.36 }{ 14.14 } = 5 ; ;

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{ 13.07 }{ 14.14 } ) = 67° 30' ; ; sin beta = fraction{ b }{ c } ; ; beta = arcsin( fraction{ b }{ c } ) = arcsin( fraction{ 5.41 }{ 14.14 } ) = 22° 30' ; ; gamma = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 35.36 }{ 16.31 } = 2.17 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 13.07 }{ 2 * sin 67° 30' } = 7.07 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.41**2+2 * 14.14**2 - 13.07**2 } }{ 2 } = 8.483 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 14.14**2+2 * 13.07**2 - 5.41**2 } }{ 2 } = 13.343 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 5.41**2+2 * 13.07**2 - 14.14**2 } }{ 2 } = 7.071 ; ;
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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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