Triangle calculator AAS

Please enter two angles and one opposite side
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Right scalene triangle.

Sides: a = 312   b = 54.17882314321   c = 307.266001894

Area: T = 8323.402220797
Perimeter: p = 673.4388250372
Semiperimeter: s = 336.7199125186

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 10° = 0.17545329252 rad
Angle ∠ C = γ = 80° = 1.39662634016 rad

Height: ha = 53.35551423588
Height: hb = 307.266001894
Height: hc = 54.17882314321

Median: ma = 156
Median: mb = 308.4521842966
Median: mc = 162.9033224556

Inradius: r = 24.71991251859
Circumradius: R = 156

Vertex coordinates: A[307.266001894; 0] B[0; 0] C[307.266001894; 54.17882314321]
Centroid: CG[204.8440012626; 18.05994104774]
Coordinates of the circumscribed circle: U[153.633000947; 27.0899115716]
Coordinates of the inscribed circle: I[282.5410893754; 24.71991251859]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 170° = 0.17545329252 rad
∠ C' = γ' = 100° = 1.39662634016 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 10° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 10° = 80° ; ;

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

a = 312 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 312 * fraction{ sin 10° }{ sin 90° } = 54.18 ; ;

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

 fraction{ c }{ a } = fraction{ sin gamma }{ sin alpha } ; ; ; ; c = a * fraction{ sin gamma }{ sin alpha } ; ; ; ; c = 312 * fraction{ sin 80° }{ sin 90° } = 307.26 ; ;
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 = 312 ; ; b = 54.18 ; ; c = 307.26 ; ;

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

p = a+b+c = 312+54.18+307.26 = 673.44 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 673.44 }{ 2 } = 336.72 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 336.72 * (336.72-312)(336.72-54.18)(336.72-307.26) } ; ; T = sqrt{ 69279024.32 } = 8323.4 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8323.4 }{ 312 } = 53.36 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8323.4 }{ 54.18 } = 307.26 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8323.4 }{ 307.26 } = 54.18 ; ;

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{ 54.18**2+307.26**2-312**2 }{ 2 * 54.18 * 307.26 } ) = 90° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 312**2+307.26**2-54.18**2 }{ 2 * 312 * 307.26 } ) = 10° ; ;
 gamma = 180° - alpha - beta = 180° - 90° - 10° = 80° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8323.4 }{ 336.72 } = 24.72 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 312 }{ 2 * sin 90° } = 156 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 54.18**2+2 * 307.26**2 - 312**2 } }{ 2 } = 156 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 307.26**2+2 * 312**2 - 54.18**2 } }{ 2 } = 308.452 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 54.18**2+2 * 312**2 - 307.26**2 } }{ 2 } = 162.903 ; ;
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