Triangle calculator ASA

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

Sides: a = 288.4310727886   b = 186.2439820993   c = 146

Area: T = 11774.05443809
Perimeter: p = 620.6710548879
Semiperimeter: s = 310.335527444

Angle ∠ A = α = 120° = 2.09443951024 rad
Angle ∠ B = β = 34° = 0.59334119457 rad
Angle ∠ C = γ = 26° = 0.45437856055 rad

Height: ha = 81.64221639067
Height: hb = 126.4439708952
Height: hc = 161.2888416177

Median: ma = 84.86879224717
Median: mb = 208.7654998655
Median: mc = 231.5387854046

Inradius: r = 37.9439787548
Circumradius: R = 166.5265558388

Vertex coordinates: A[146; 0] B[0; 0] C[239.1219910497; 161.2888416177]
Centroid: CG[128.3733303499; 53.76328053922]
Coordinates of the circumscribed circle: U[73; 149.6722180435]
Coordinates of the inscribed circle: I[124.0955453446; 37.9439787548]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 60° = 2.09443951024 rad
∠ B' = β' = 146° = 0.59334119457 rad
∠ C' = γ' = 154° = 0.45437856055 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 120° ; ; beta = 34° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 120° - 34° = 26° ; ;

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

c = 146 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 146 * fraction{ sin(120° ) }{ sin (26° ) } = 288.43 ; ;

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 = 146 * fraction{ sin(34° ) }{ sin (26° ) } = 186.24 ; ;


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 = 288.43 ; ; b = 186.24 ; ; c = 146 ; ;

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

p = a+b+c = 288.43+186.24+146 = 620.67 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 620.67 }{ 2 } = 310.34 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 310.34 * (310.34-288.43)(310.34-186.24)(310.34-146) } ; ; T = sqrt{ 138628356.56 } = 11774.05 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 11774.05 }{ 288.43 } = 81.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 11774.05 }{ 186.24 } = 126.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 11774.05 }{ 146 } = 161.29 ; ;

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{ 288.43**2-186.24**2-146**2 }{ 2 * 186.24 * 146 } ) = 120° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 186.24**2-288.43**2-146**2 }{ 2 * 288.43 * 146 } ) = 34° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 146**2-288.43**2-186.24**2 }{ 2 * 186.24 * 288.43 } ) = 26° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 11774.05 }{ 310.34 } = 37.94 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 288.43 }{ 2 * sin 120° } = 166.53 ; ;

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