Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 15.2   b = 13.80658534981   c = 23.4810532122

Area: T = 99.78991166859
Perimeter: p = 52.48663856201
Semiperimeter: s = 26.24331928101

Angle ∠ A = α = 38° = 0.66332251158 rad
Angle ∠ B = β = 34° = 0.59334119457 rad
Angle ∠ C = γ = 108° = 1.88549555922 rad

Height: ha = 13.13301469324
Height: hb = 14.45660590477
Height: hc = 8.54997321328

Median: ma = 17.69876973014
Median: mb = 18.53547591477
Median: mc = 8.54332399136

Inradius: r = 3.80224762234
Circumradius: R = 12.34444462657

Vertex coordinates: A[23.4810532122; 0] B[0; 0] C[12.60113711028; 8.54997321328]
Centroid: CG[12.0277301075; 2.83332440443]
Coordinates of the circumscribed circle: U[11.7440266061; -3.81546436822]
Coordinates of the inscribed circle: I[12.4377339312; 3.80224762234]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142° = 0.66332251158 rad
∠ B' = β' = 146° = 0.59334119457 rad
∠ C' = γ' = 72° = 1.88549555922 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 38° ; ; beta = 34° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 38° - 34° = 108° ; ;

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

a = 15.2 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 15.2 * fraction{ sin(34° ) }{ sin (38° ) } = 13.81 ; ;

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 = 15.2 * fraction{ sin(108° ) }{ sin (38° ) } = 23.48 ; ;


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 = 15.2 ; ; b = 13.81 ; ; c = 23.48 ; ;

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

p = a+b+c = 15.2+13.81+23.48 = 52.49 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 52.49 }{ 2 } = 26.24 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 26.24 * (26.24-15.2)(26.24-13.81)(26.24-23.48) } ; ; T = sqrt{ 9957.87 } = 99.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 99.79 }{ 15.2 } = 13.13 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 99.79 }{ 13.81 } = 14.46 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 99.79 }{ 23.48 } = 8.5 ; ;

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{ 15.2**2-13.81**2-23.48**2 }{ 2 * 13.81 * 23.48 } ) = 38° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 13.81**2-15.2**2-23.48**2 }{ 2 * 15.2 * 23.48 } ) = 34° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23.48**2-15.2**2-13.81**2 }{ 2 * 13.81 * 15.2 } ) = 108° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 99.79 }{ 26.24 } = 3.8 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 15.2 }{ 2 * sin 38° } = 12.34 ; ;




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