Triangle calculator

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

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

Acute scalene triangle.

Sides: a = 146.358758202   b = 110   c = 178.4254756503

Area: T = 8038.635522225
Perimeter: p = 434.7822338523
Semiperimeter: s = 217.3911169262

Angle ∠ A = α = 55° = 0.96599310886 rad
Angle ∠ B = β = 38° = 0.66332251158 rad
Angle ∠ C = γ = 87° = 1.51884364492 rad

Height: ha = 109.8499248823
Height: hb = 146.1577004041
Height: hc = 90.10767248718

Median: ma = 128.8989725784
Median: mb = 153.6332573935
Median: mc = 93.8165896703

Inradius: r = 36.97877450002
Circumradius: R = 89.33548085016

Vertex coordinates: A[178.4254756503; 0] B[0; 0] C[115.3311348504; 90.10767248718]
Centroid: CG[97.91987016689; 30.03655749573]
Coordinates of the circumscribed circle: U[89.21223782513; 4.67554226287]
Coordinates of the inscribed circle: I[107.3911169261; 36.97877450002]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125° = 0.96599310886 rad
∠ B' = β' = 142° = 0.66332251158 rad
∠ C' = γ' = 93° = 1.51884364492 rad

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How did we calculate this triangle?

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

b = 110 ; ; alpha = 55° ; ; beta = 38° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 55 ° - 38 ° = 87 ° ; ;

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

 fraction{ a }{ b } = fraction{ sin alpha }{ sin beta } ; ; ; ; a = b * fraction{ sin alpha }{ sin beta } ; ; ; ; a = 110 * fraction{ sin 55° }{ sin 38° } = 146.36 ; ;

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

c**2 = b**2+a**2 - 2ba cos gamma ; ; c = sqrt{ b**2+a**2 - 2ba cos gamma } ; ; c = sqrt{ 110**2+146.36**2 - 2 * 110 * 146.36 * cos(87° ) } ; ; c = 178.42 ; ;


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 = 146.36 ; ; b = 110 ; ; c = 178.42 ; ;

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

p = a+b+c = 146.36+110+178.42 = 434.78 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 434.78 }{ 2 } = 217.39 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 217.39 * (217.39-146.36)(217.39-110)(217.39-178.42) } ; ; T = sqrt{ 64619656.24 } = 8038.64 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8038.64 }{ 146.36 } = 109.85 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8038.64 }{ 110 } = 146.16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8038.64 }{ 178.42 } = 90.11 ; ;

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{ 110**2+178.42**2-146.36**2 }{ 2 * 110 * 178.42 } ) = 55° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 146.36**2+178.42**2-110**2 }{ 2 * 146.36 * 178.42 } ) = 38° ; ; gamma = arccos( fraction{ a**2+b**2-c**2 }{ 2ab } ) = arccos( fraction{ 146.36**2+110**2-178.42**2 }{ 2 * 146.36 * 110 } ) = 87° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8038.64 }{ 217.39 } = 36.98 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 146.36 }{ 2 * sin 55° } = 89.33 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 110**2+2 * 178.42**2 - 146.36**2 } }{ 2 } = 128.89 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 178.42**2+2 * 146.36**2 - 110**2 } }{ 2 } = 153.633 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 110**2+2 * 146.36**2 - 178.42**2 } }{ 2 } = 93.816 ; ;
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