Triangle calculator ASA

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

Sides: a = 163.2999316186   b = 59.77216981445   c = 200

Area: T = 4226.49773081
Perimeter: p = 423.071101433
Semiperimeter: s = 211.5365507165

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: ha = 51.76438090205
Height: hb = 141.4211356237
Height: hc = 42.2654973081

Median: ma = 122.9622031875
Median: mb = 180.1121546988
Median: mc = 71.5521808383

Inradius: r = 19.98800845009
Circumradius: R = 115.4770053838

Vertex coordinates: A[200; 0] B[0; 0] C[157.7355026919; 42.2654973081]
Centroid: CG[119.2455008973; 14.08883243603]
Coordinates of the circumscribed circle: U[100; -57.7355026919]
Coordinates of the inscribed circle: I[151.764380902; 19.98800845009]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 60° = 2.09443951024 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 45° ; ; beta = 15° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 45° - 15° = 120° ; ;

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

c = 200 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 200 * fraction{ sin(45° ) }{ sin (120° ) } = 163.3 ; ;

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 = 200 * fraction{ sin(15° ) }{ sin (120° ) } = 59.77 ; ;


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 = 163.3 ; ; b = 59.77 ; ; c = 200 ; ;

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

p = a+b+c = 163.3+59.77+200 = 423.07 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 423.07 }{ 2 } = 211.54 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 211.54 * (211.54-163.3)(211.54-59.77)(211.54-200) } ; ; T = sqrt{ 17863279.5 } = 4226.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 4226.5 }{ 163.3 } = 51.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 4226.5 }{ 59.77 } = 141.42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 4226.5 }{ 200 } = 42.26 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 163.3**2-59.77**2-200**2 }{ 2 * 59.77 * 200 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 59.77**2-163.3**2-200**2 }{ 2 * 163.3 * 200 } ) = 15° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 200**2-163.3**2-59.77**2 }{ 2 * 59.77 * 163.3 } ) = 120° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 4226.5 }{ 211.54 } = 19.98 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 163.3 }{ 2 * sin 45° } = 115.47 ; ;




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