Right triangle calculator (A,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 = 13.40656277746   b = 109.188007668   c = 110

Area: T = 731.8143734189
Perimeter: p = 232.5865704455
Semiperimeter: s = 116.2932852228

Angle ∠ A = α = 7° = 0.12221730476 rad
Angle ∠ B = β = 83° = 1.44986232792 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 109.188007668
Height: hb = 13.40656277746
Height: hc = 13.3065704258

Median: ma = 109.3865633691
Median: mb = 56.2121948392
Median: mc = 55

Inradius: r = 6.29328522276
Circumradius: R = 55

Vertex coordinates: A[110; 0] B[0; 0] C[1.63437350548; 13.3065704258]
Centroid: CG[37.21112450183; 4.43552347527]
Coordinates of the circumscribed circle: U[55; 0]
Coordinates of the inscribed circle: I[7.1132775547; 6.29328522276]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 173° = 0.12221730476 rad
∠ B' = β' = 97° = 1.44986232792 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: hypotenuse c angle α

c = 110 ; ; alpha = 7° ; ;

2. From angle α we calculate angle β:

 alpha + beta + 90° = 180° ; ; beta = 90° - alpha = 90° - 7 ° = 83 ° ; ;

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

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(7 ° ) = 13.406 ; ;

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{ 110**2 - 13.406**2 } = 109.18 ; ;


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 = 13.41 ; ; b = 109.18 ; ; c = 110 ; ;

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

p = a+b+c = 13.41+109.18+110 = 232.59 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 232.59 }{ 2 } = 116.29 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 116.29 * (116.29-13.41)(116.29-109.18)(116.29-110) } ; ; T = sqrt{ 535551.34 } = 731.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 731.81 }{ 13.41 } = 109.18 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 731.81 }{ 109.18 } = 13.41 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 731.81 }{ 110 } = 13.31 ; ;

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{ 13.41**2-109.18**2-110**2 }{ 2 * 109.18 * 110 } ) = 7° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 109.18**2-13.41**2-110**2 }{ 2 * 13.41 * 110 } ) = 83° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 110**2-13.41**2-109.18**2 }{ 2 * 109.18 * 13.41 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 731.81 }{ 116.29 } = 6.29 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13.41 }{ 2 * sin 7° } = 55 ; ;
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