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.

Obtuse scalene triangle.

Sides: a = 46.06597304655   b = 41   c = 74.67702412564

Area: T = 833.7010726916
Perimeter: p = 161.7329971722
Semiperimeter: s = 80.86549858609

Angle ∠ A = α = 33° = 0.57659586532 rad
Angle ∠ B = β = 29° = 0.50661454831 rad
Angle ∠ C = γ = 118° = 2.05994885174 rad

Height: ha = 36.20108513072
Height: hb = 40.66883281422
Height: hc = 22.33302004356

Median: ma = 55.65992110257
Median: mb = 58.55218731547
Median: mc = 22.52441681967

Inradius: r = 10.31097863437
Circumradius: R = 42.28546394624

Vertex coordinates: A[74.67702412564; 0] B[0; 0] C[40.28547479706; 22.33302004356]
Centroid: CG[38.31883297423; 7.44334001452]
Coordinates of the circumscribed circle: U[37.33551206282; -19.85114357702]
Coordinates of the inscribed circle: I[39.86549858609; 10.31097863437]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147° = 0.57659586532 rad
∠ B' = β' = 151° = 0.50661454831 rad
∠ C' = γ' = 62° = 2.05994885174 rad

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How did we calculate this triangle?

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

b = 41 ; ; alpha = 33° ; ; beta = 29° ; ;

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

 alpha + beta + gamma = 180° ; ; gamma = 180° - alpha - beta = 180° - 33 ° - 29 ° = 118 ° ; ;

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 = 41 * fraction{ sin 33° }{ sin 29° } = 46.06 ; ;

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{ 41**2+46.06**2 - 2 * 41 * 46.06 * cos 118° } ; ; c = 74.67 ; ;
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 = 46.06 ; ; b = 41 ; ; c = 74.67 ; ;

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

p = a+b+c = 46.06+41+74.67 = 161.73 ; ;

6. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 161.73 }{ 2 } = 80.86 ; ;

7. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 80.86 * (80.86-46.06)(80.86-41)(80.86-74.67) } ; ; T = sqrt{ 695056.9 } = 833.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 833.7 }{ 46.06 } = 36.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 833.7 }{ 41 } = 40.67 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 833.7 }{ 74.67 } = 22.33 ; ;

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{ 41**2+74.67**2-46.06**2 }{ 2 * 41 * 74.67 } ) = 33° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 46.06**2+74.67**2-41**2 }{ 2 * 46.06 * 74.67 } ) = 29° ; ; gamma = 180° - alpha - beta = 180° - 33° - 29° = 118° ; ;

10. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 833.7 }{ 80.86 } = 10.31 ; ;

11. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 46.06 }{ 2 * sin 33° } = 42.28 ; ;

12. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 41**2+2 * 74.67**2 - 46.06**2 } }{ 2 } = 55.659 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 74.67**2+2 * 46.06**2 - 41**2 } }{ 2 } = 58.552 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 41**2+2 * 46.06**2 - 74.67**2 } }{ 2 } = 22.524 ; ;
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