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 α.

Right scalene triangle.

Sides: a = 27.25216415365   b = 92   c = 95.95112999726

Area: T = 1253.576551068
Perimeter: p = 215.2032941509
Semiperimeter: s = 107.6011470755

Angle ∠ A = α = 16.5° = 16°30' = 0.28879793266 rad
Angle ∠ B = β = 73.5° = 73°30' = 1.28328170002 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 92
Height: hb = 27.25216415365
Height: hc = 26.12994117128

Median: ma = 93.00435644027
Median: mb = 53.46663629438
Median: mc = 47.97656499863

Inradius: r = 11.65501707819
Circumradius: R = 47.97656499863

Vertex coordinates: A[95.95112999726; 0] B[0; 0] C[7.74398843647; 26.12994117128]
Centroid: CG[34.56437281125; 8.71098039043]
Coordinates of the circumscribed circle: U[47.97656499863; -0]
Coordinates of the inscribed circle: I[15.60114707546; 11.65501707819]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 163.5° = 163°30' = 0.28879793266 rad
∠ B' = β' = 106.5° = 106°30' = 1.28328170002 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: cathetus b angle α

b = 92 ; ; alpha = 16.5° ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 16.5 ° = 73.5 ° ; ;

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

 cos alpha = b:c ; ; c = b/ cos alpha = b/ cos(16.5 ° ) = 95.951 ; ;

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

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(16.5 ° ) = 27.252 ; ;


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 = 27.25 ; ; b = 92 ; ; c = 95.95 ; ;

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

p = a+b+c = 27.25+92+95.95 = 215.2 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 215.2 }{ 2 } = 107.6 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 107.6 * (107.6-27.25)(107.6-92)(107.6-95.95) } ; ; T = sqrt{ 1571451.56 } = 1253.58 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1253.58 }{ 27.25 } = 92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1253.58 }{ 92 } = 27.25 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1253.58 }{ 95.95 } = 26.13 ; ;

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{ 27.25**2-92**2-95.95**2 }{ 2 * 92 * 95.95 } ) = 16° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 92**2-27.25**2-95.95**2 }{ 2 * 27.25 * 95.95 } ) = 73° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 95.95**2-27.25**2-92**2 }{ 2 * 92 * 27.25 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1253.58 }{ 107.6 } = 11.65 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 27.25 }{ 2 * sin 16° 30' } = 47.98 ; ;
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