Triangle calculator

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

You have entered side a, angle α and angle γ.

Acute scalene triangle.

Sides: a = 36   b = 37.73768234214   c = 41.25882417879

Area: T = 629.8021521181
Perimeter: p = 114.9955065209
Semiperimeter: s = 57.49875326046

Angle ∠ A = α = 54° = 0.94224777961 rad
Angle ∠ B = β = 58° = 1.01222909662 rad
Angle ∠ C = γ = 68° = 1.18768238914 rad

Height: ha = 34.9898973399
Height: hb = 33.37986187644
Height: hc = 30.53297314616

Median: ma = 35.20216360228
Median: mb = 33.8109825454
Median: mc = 30.56991558947

Inradius: r = 10.95435399634
Circumradius: R = 22.2499223595

Vertex coordinates: A[41.25882417879; 0] B[0; 0] C[19.07770935124; 30.53297314616]
Centroid: CG[20.11217784334; 10.17765771539]
Coordinates of the circumscribed circle: U[20.62991208939; 8.33547058571]
Coordinates of the inscribed circle: I[19.76107091832; 10.95435399634]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126° = 0.94224777961 rad
∠ B' = β' = 122° = 1.01222909662 rad
∠ C' = γ' = 112° = 1.18768238914 rad

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How did we calculate this triangle?

1. Input data entered: side a, angle α and angle γ.

a = 36 ; ; alpha = 54° ; ; gamma = 68° ; ;

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

 alpha + gamma + beta = 180° ; ; beta = 180° - alpha - gamma = 180° - 54 ° - 68 ° = 58 ° ; ;

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

 fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 36 * fraction{ sin 58° }{ sin 54° } = 37.74 ; ;

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

 fraction{ c }{ a } = fraction{ sin gamma }{ sin alpha } ; ; ; ; c = a * fraction{ sin gamma }{ sin alpha } ; ; ; ; c = 36 * fraction{ sin 68° }{ sin 54° } = 41.26 ; ;


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 = 36 ; ; b = 37.74 ; ; c = 41.26 ; ;

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

p = a+b+c = 36+37.74+41.26 = 115 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 115 }{ 2 } = 57.5 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 57.5 * (57.5-36)(57.5-37.74)(57.5-41.26) } ; ; T = sqrt{ 396649.96 } = 629.8 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 629.8 }{ 36 } = 34.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 629.8 }{ 37.74 } = 33.38 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 629.8 }{ 41.26 } = 30.53 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 37.74**2+41.26**2-36**2 }{ 2 * 37.74 * 41.26 } ) = 54° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 36**2+41.26**2-37.74**2 }{ 2 * 36 * 41.26 } ) = 58° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 36**2+37.74**2-41.26**2 }{ 2 * 36 * 37.74 } ) = 68° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 629.8 }{ 57.5 } = 10.95 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 36 }{ 2 * sin 54° } = 22.25 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 37.74**2+2 * 41.26**2 - 36**2 } }{ 2 } = 35.202 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 41.26**2+2 * 36**2 - 37.74**2 } }{ 2 } = 33.81 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 37.74**2+2 * 36**2 - 41.26**2 } }{ 2 } = 30.569 ; ;
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