Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
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Obtuse scalene triangle.

Sides: a = 29.87992351578   b = 413.7411167802   c = 405

Area: T = 5844.3787794
Perimeter: p = 848.622040296
Semiperimeter: s = 424.311020148

Angle ∠ A = α = 4° = 0.07698131701 rad
Angle ∠ B = β = 105° = 1.83325957146 rad
Angle ∠ C = γ = 71° = 1.23991837689 rad

Height: ha = 391.2199959647
Height: hb = 28.25113718664
Height: hc = 28.86111249087

Median: ma = 409.1211234836
Median: mb = 199.157683735
Median: mc = 212.2055116135

Inradius: r = 13.77438328553
Circumradius: R = 214.168818794

Vertex coordinates: A[405; 0] B[0; 0] C[-7.73333151119; 28.86111249087]
Centroid: CG[132.4222228296; 9.62203749696]
Coordinates of the circumscribed circle: U[202.5; 69.72663416912]
Coordinates of the inscribed circle: I[10.56990336778; 13.77438328553]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 176° = 0.07698131701 rad
∠ B' = β' = 75° = 1.83325957146 rad
∠ C' = γ' = 109° = 1.23991837689 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 4° ; ; beta = 105° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 4° - 105° = 71° ; ;

2. By using the law of sines, we calculate unknown side a

c = 405 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 405 * fraction{ sin 4° }{ sin 71° } = 29.88 ; ;

3. By using the law of sines, we calculate last unknown side b

 fraction{ b }{ c } = fraction{ sin beta }{ sin gamma } ; ; ; ; b = c * fraction{ sin beta }{ sin gamma } ; ; ; ; b = 405 * fraction{ sin 105° }{ sin 71° } = 413.74 ; ;
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 = 29.88 ; ; b = 413.74 ; ; c = 405 ; ;

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

p = a+b+c = 29.88+413.74+405 = 848.62 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 848.62 }{ 2 } = 424.31 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 424.31 * (424.31-29.88)(424.31-413.74)(424.31-405) } ; ; T = sqrt{ 34156751.8 } = 5844.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5844.38 }{ 29.88 } = 391.2 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5844.38 }{ 413.74 } = 28.25 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5844.38 }{ 405 } = 28.86 ; ;

8. 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{ 413.74**2+405**2-29.88**2 }{ 2 * 413.74 * 405 } ) = 4° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 29.88**2+405**2-413.74**2 }{ 2 * 29.88 * 405 } ) = 105° ; ; gamma = 180° - alpha - beta = 180° - 4° - 105° = 71° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5844.38 }{ 424.31 } = 13.77 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 29.88 }{ 2 * sin 4° } = 214.17 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 413.74**2+2 * 405**2 - 29.88**2 } }{ 2 } = 409.121 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 405**2+2 * 29.88**2 - 413.74**2 } }{ 2 } = 199.157 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 413.74**2+2 * 29.88**2 - 405**2 } }{ 2 } = 212.205 ; ;
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