Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 112   b = 171.5943955259   c = 220.5976936675

Area: T = 9463.275522049
Perimeter: p = 504.1910891933
Semiperimeter: s = 252.0955445967

Angle ∠ A = α = 30° = 0.52435987756 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 100° = 1.7455329252 rad

Height: ha = 168.9877057509
Height: hb = 110.2988468337
Height: hc = 85.79769776293

Median: ma = 189.5219779906
Median: mb = 152.4554527203
Median: mc = 93.96600480156

Inradius: r = 37.53884616101
Circumradius: R = 112

Vertex coordinates: A[220.5976936675; 0] B[0; 0] C[71.99222122849; 85.79769776293]
Centroid: CG[97.53297163199; 28.59989925431]
Coordinates of the circumscribed circle: U[110.2988468337; -19.44985958987]
Coordinates of the inscribed circle: I[80.5011490708; 37.53884616101]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 150° = 0.52435987756 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 80° = 1.7455329252 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 30° ; ; beta = 50° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 30° - 50° = 100° ; ;

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

a = 112 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 112 * fraction{ sin(50° ) }{ sin (30° ) } = 171.59 ; ;

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 = 112 * fraction{ sin(100° ) }{ sin (30° ) } = 220.6 ; ;


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 = 112 ; ; b = 171.59 ; ; c = 220.6 ; ;

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

p = a+b+c = 112+171.59+220.6 = 504.19 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 504.19 }{ 2 } = 252.1 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 252.1 * (252.1-112)(252.1-171.59)(252.1-220.6) } ; ; T = sqrt{ 89553577.9 } = 9463.28 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9463.28 }{ 112 } = 168.99 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9463.28 }{ 171.59 } = 110.3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9463.28 }{ 220.6 } = 85.8 ; ;

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{ 112**2-171.59**2-220.6**2 }{ 2 * 171.59 * 220.6 } ) = 30° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 171.59**2-112**2-220.6**2 }{ 2 * 112 * 220.6 } ) = 50° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 220.6**2-112**2-171.59**2 }{ 2 * 171.59 * 112 } ) = 100° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9463.28 }{ 252.1 } = 37.54 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 112 }{ 2 * sin 30° } = 112 ; ;




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