Right triangle calculator (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 α.

Right scalene triangle.

Sides: a = 47.70992713321   b = 22.62200227444   c = 52.8

Area: T = 539.5922401325
Perimeter: p = 123.1299294077
Semiperimeter: s = 61.56546470383

Angle ∠ A = α = 64.63333333333° = 64°38' = 1.12880644732 rad
Angle ∠ B = β = 25.36766666667° = 25°22' = 0.44327318536 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 22.62200227444
Height: hb = 47.70992713321
Height: hc = 20.43991061108

Median: ma = 32.87441398628
Median: mb = 49.03215299403
Median: mc = 26.4

Inradius: r = 8.76546470383
Circumradius: R = 26.4

Vertex coordinates: A[52.8; 0] B[0; 0] C[43.10993668758; 20.43991061108]
Centroid: CG[31.97697889586; 6.81330353703]
Coordinates of the circumscribed circle: U[26.4; 0]
Coordinates of the inscribed circle: I[38.94546242939; 8.76546470383]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 115.3676666667° = 115°22' = 1.12880644732 rad
∠ B' = β' = 154.6333333333° = 154°38' = 0.44327318536 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 α

c = 52.8 ; ; alpha = 64.633° ; ;

2. From angle α we calculate angle β:

alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 64.633 ° = 25.367 ° ; ;

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

sin alpha = a:c ; ; a = c * sin alpha = 52.8 * sin(64.633 ° ) = 47.709 ; ;

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{ 52.8**2 - 47.709**2 } = 22.62 ; ;


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 = 47.71 ; ; b = 22.62 ; ; c = 52.8 ; ;

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

p = a+b+c = 47.71+22.62+52.8 = 123.13 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 123.13 }{ 2 } = 61.56 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 61.56 * (61.56-47.71)(61.56-22.62)(61.56-52.8) } ; ; T = sqrt{ 291159.96 } = 539.59 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 539.59 }{ 47.71 } = 22.62 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 539.59 }{ 22.62 } = 47.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 539.59 }{ 52.8 } = 20.44 ; ;

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{ 47.71**2-22.62**2-52.8**2 }{ 2 * 22.62 * 52.8 } ) = 64° 38' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22.62**2-47.71**2-52.8**2 }{ 2 * 47.71 * 52.8 } ) = 25° 22' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 52.8**2-47.71**2-22.62**2 }{ 2 * 22.62 * 47.71 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 539.59 }{ 61.56 } = 8.76 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 47.71 }{ 2 * sin 64° 38' } = 26.4 ; ;
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