Triangle calculator ASA

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

Sides: a = 141.9299455624   b = 84.51659493122   c = 200

Area: T = 5086.29767992
Perimeter: p = 426.4455404936
Semiperimeter: s = 213.2232702468

Angle ∠ A = α = 37° = 0.64657718232 rad
Angle ∠ B = β = 21° = 0.36765191429 rad
Angle ∠ C = γ = 122° = 2.12993016874 rad

Height: ha = 71.67435899091
Height: hb = 120.363300463
Height: hc = 50.8632967992

Median: ma = 136.1455070607
Median: mb = 168.1855162142
Median: mc = 60.36110638651

Inradius: r = 23.85443867061
Circumradius: R = 117.9187840336

Vertex coordinates: A[200; 0] B[0; 0] C[132.5032561714; 50.8632967992]
Centroid: CG[110.8344187238; 16.9544322664]
Coordinates of the circumscribed circle: U[100; -62.48769351909]
Coordinates of the inscribed circle: I[128.7076753156; 23.85443867061]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 143° = 0.64657718232 rad
∠ B' = β' = 159° = 0.36765191429 rad
∠ C' = γ' = 58° = 2.12993016874 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 37° ; ; beta = 21° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 37° - 21° = 122° ; ;

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 37° }{ sin 122° } = 141.93 ; ;

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 21° }{ sin 122° } = 84.52 ; ;
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 = 141.93 ; ; b = 84.52 ; ; c = 200 ; ;

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

p = a+b+c = 141.93+84.52+200 = 426.45 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 426.45 }{ 2 } = 213.22 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 213.22 * (213.22-141.93)(213.22-84.52)(213.22-200) } ; ; T = sqrt{ 25870415.13 } = 5086.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5086.3 }{ 141.93 } = 71.67 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5086.3 }{ 84.52 } = 120.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5086.3 }{ 200 } = 50.86 ; ;

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{ 84.52**2+200**2-141.93**2 }{ 2 * 84.52 * 200 } ) = 37° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 141.93**2+200**2-84.52**2 }{ 2 * 141.93 * 200 } ) = 21° ; ;
 gamma = 180° - alpha - beta = 180° - 37° - 21° = 122° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5086.3 }{ 213.22 } = 23.85 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 141.93 }{ 2 * sin 37° } = 117.92 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 84.52**2+2 * 200**2 - 141.93**2 } }{ 2 } = 136.145 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 200**2+2 * 141.93**2 - 84.52**2 } }{ 2 } = 168.185 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 84.52**2+2 * 141.93**2 - 200**2 } }{ 2 } = 60.361 ; ;
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