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 height h

 beta = 22.5° ; ; hc = 5 ; ;

2. From angle β we calculate angle α:

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

3. From and angle α we calculate hypotenuse c:

 sin alpha = a:c ; ; c = a/ sin alpha = a/ 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 ; ;

4. 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 ; ;

5. Semiperimeter of the triangle

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

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 16.31 * (16.31-13.07)(16.31-5.41)(16.31-14.14) } ; ; T = sqrt{ 1250 } = 35.36 ; ;

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

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

8. 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{ 13.07**2-5.41**2-14.14**2 }{ 2 * 5.41 * 14.14 } ) = 67° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 5.41**2-13.07**2-14.14**2 }{ 2 * 13.07 * 14.14 } ) = 22° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 14.14**2-13.07**2-5.41**2 }{ 2 * 5.41 * 13.07 } ) = 90° ; ;

9. Inradius

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

10. Circumradius

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