Right triangle calculator (B,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 β.

Right scalene triangle.

Sides: a = 76.60444443119   b = 64.27987609687   c = 100

Area: T = 2462.019938253
Perimeter: p = 240.8833205281
Semiperimeter: s = 120.442160264

Angle ∠ A = α = 50° = 0.8732664626 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 64.27987609687
Height: hb = 76.60444443119
Height: hc = 49.24403876506

Median: ma = 74.82552586614
Median: mb = 83.07333451009
Median: mc = 50

Inradius: r = 20.44216026403
Circumradius: R = 50

Vertex coordinates: A[100; 0] B[0; 0] C[58.68224088833; 49.24403876506]
Centroid: CG[52.89441362944; 16.41334625502]
Coordinates of the circumscribed circle: U[50; -0]
Coordinates of the inscribed circle: I[56.16328416716; 20.44216026403]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130° = 0.8732664626 rad
∠ B' = β' = 140° = 0.69881317008 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Input data entered: hypotenuse c angle β

c = 100 ; ; beta = 40° ; ;

2. From angle β we calculate angle α:

 alpha + beta + 90° = 180° ; ; alpha = 90° - beta = 90° - 40 ° = 50 ° ; ;

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

 sin alpha = a:c ; ; a = c * sin alpha = c * sin(50 ° ) = 76.604 ; ;

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{ 100**2 - 76.604**2 } = 64.279 ; ;


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 = 76.6 ; ; b = 64.28 ; ; c = 100 ; ;

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

p = a+b+c = 76.6+64.28+100 = 240.88 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 240.88 }{ 2 } = 120.44 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 120.44 * (120.44-76.6)(120.44-64.28)(120.44-100) } ; ; T = sqrt{ 6061539.44 } = 2462.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2462.02 }{ 76.6 } = 64.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2462.02 }{ 64.28 } = 76.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2462.02 }{ 100 } = 49.24 ; ;

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{ 76.6**2-64.28**2-100**2 }{ 2 * 64.28 * 100 } ) = 50° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 64.28**2-76.6**2-100**2 }{ 2 * 76.6 * 100 } ) = 40° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 100**2-76.6**2-64.28**2 }{ 2 * 64.28 * 76.6 } ) = 90° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2462.02 }{ 120.44 } = 20.44 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 76.6 }{ 2 * sin 50° } = 50 ; ;
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