Triangle calculator

Please enter what you know about the triangle:
Symbols definition of ABC triangle

You have entered side c, angle α and angle β.

Obtuse scalene triangle.

Sides: a = 16.24659848116   b = 42.53325404176   c = 50

Area: T = 328.5821945075
Perimeter: p = 108.7798525229
Semiperimeter: s = 54.38992626146

Angle ∠ A = α = 18° = 0.31441592654 rad
Angle ∠ B = β = 54° = 0.94224777961 rad
Angle ∠ C = γ = 108° = 1.88549555922 rad

Height: ha = 40.45108497187
Height: hb = 15.45108497187
Height: hc = 13.1433277803

Median: ma = 45.77003883087
Median: mb = 30.49111751603
Median: mc = 20.28548344444

Inradius: r = 6.0411301707
Circumradius: R = 26.2876555606

Vertex coordinates: A[50; 0] B[0; 0] C[9.54991502813; 13.1433277803]
Centroid: CG[19.85497167604; 4.3811092601]
Coordinates of the circumscribed circle: U[25; -8.12329924058]
Coordinates of the inscribed circle: I[11.8576722197; 6.0411301707]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162° = 0.31441592654 rad
∠ B' = β' = 126° = 0.94224777961 rad
∠ C' = γ' = 72° = 1.88549555922 rad

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How did we calculate this triangle?

1. Input data entered: side c, angle α and angle β.

c = 50 ; ; alpha = 18° ; ; beta = 54° ; ;

2. From angle α and angle β we calculate γ:

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 18 ° - 54 ° = 108 ° ; ;

3. From angle α, angle γ and side c we calculate a - By using the law of sines, we calculate unknown side a:

 fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 50 * fraction{ sin(18° ) }{ sin (108° ) } = 16.25 ; ;

4. Calculation of the third side b of the triangle using a Law of Cosines

b**2 = a**2+c**2 - 2ac cos beta ; ; b = sqrt{ a**2+c**2 - 2ac cos beta } ; ; b = sqrt{ 16.25**2+50**2 - 2 * 16.25 * 50 * cos(54° ) } ; ; b = 42.53 ; ;


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 = 16.25 ; ; b = 42.53 ; ; c = 50 ; ;

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

p = a+b+c = 16.25+42.53+50 = 108.78 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 108.78 }{ 2 } = 54.39 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 54.39 * (54.39-16.25)(54.39-42.53)(54.39-50) } ; ; T = sqrt{ 107966.09 } = 328.58 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 328.58 }{ 16.25 } = 40.45 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 328.58 }{ 42.53 } = 15.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 328.58 }{ 50 } = 13.14 ; ;

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{ 16.25**2-42.53**2-50**2 }{ 2 * 42.53 * 50 } ) = 18° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 42.53**2-16.25**2-50**2 }{ 2 * 16.25 * 50 } ) = 54° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 50**2-16.25**2-42.53**2 }{ 2 * 42.53 * 16.25 } ) = 108° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 328.58 }{ 54.39 } = 6.04 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 16.25 }{ 2 * sin 18° } = 26.29 ; ;




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