Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 100   b = 187.9398524157   c = 253.2098888624

Area: T = 8137.977681349
Perimeter: p = 541.1477412781
Semiperimeter: s = 270.574370639

Angle ∠ A = α = 20° = 0.34990658504 rad
Angle ∠ B = β = 40° = 0.69881317008 rad
Angle ∠ C = γ = 120° = 2.09443951024 rad

Height: ha = 162.765953627
Height: hb = 86.60325403784
Height: hc = 64.27987609687

Median: ma = 217.297660621
Median: mb = 168.0099370047
Median: mc = 81.43656132884

Inradius: r = 30.07767466361
Circumradius: R = 146.1990220008

Vertex coordinates: A[253.2098888624; 0] B[0; 0] C[76.60444443119; 64.27987609687]
Centroid: CG[109.9387777645; 21.42662536562]
Coordinates of the circumscribed circle: U[126.6044444312; -73.09551100041]
Coordinates of the inscribed circle: I[82.63551822333; 30.07767466361]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160° = 0.34990658504 rad
∠ B' = β' = 140° = 0.69881317008 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 = 20° ; ; beta = 40° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 20° - 40° = 120° ; ;

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

a = 100 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 100 * fraction{ sin 40° }{ sin 20° } = 187.94 ; ;

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 = 100 * fraction{ sin 120° }{ sin 20° } = 253.21 ; ;
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 = 100 ; ; b = 187.94 ; ; c = 253.21 ; ;

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

p = a+b+c = 100+187.94+253.21 = 541.15 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 541.15 }{ 2 } = 270.57 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 270.57 * (270.57-100)(270.57-187.94)(270.57-253.21) } ; ; T = sqrt{ 66226666.62 } = 8137.98 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 8137.98 }{ 100 } = 162.76 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 8137.98 }{ 187.94 } = 86.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 8137.98 }{ 253.21 } = 64.28 ; ;

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{ 187.94**2+253.21**2-100**2 }{ 2 * 187.94 * 253.21 } ) = 20° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 100**2+253.21**2-187.94**2 }{ 2 * 100 * 253.21 } ) = 40° ; ; gamma = 180° - alpha - beta = 180° - 20° - 40° = 120° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 8137.98 }{ 270.57 } = 30.08 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 100 }{ 2 * sin 20° } = 146.19 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 187.94**2+2 * 253.21**2 - 100**2 } }{ 2 } = 217.297 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 253.21**2+2 * 100**2 - 187.94**2 } }{ 2 } = 168.009 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 187.94**2+2 * 100**2 - 253.21**2 } }{ 2 } = 81.436 ; ;
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