Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 4.6   b = 1.49985263723   c = 4.29770876316

Area: T = 3.21876882464
Perimeter: p = 10.39656140038
Semiperimeter: s = 5.19878070019

Angle ∠ A = α = 92° = 1.60657029118 rad
Angle ∠ B = β = 19° = 0.33216125579 rad
Angle ∠ C = γ = 69° = 1.20442771839 rad

Height: ha = 1.39989948898
Height: hb = 4.29444699619
Height: hc = 1.49876135105

Median: ma = 2.25106158493
Median: mb = 4.38876059229
Median: mc = 2.66220574967

Inradius: r = 0.61990472723
Circumradius: R = 2.30114019519

Vertex coordinates: A[4.29770876316; 0] B[0; 0] C[4.34993854478; 1.49876135105]
Centroid: CG[2.88221576931; 0.49992045035]
Coordinates of the circumscribed circle: U[2.14985438158; 0.82547486986]
Coordinates of the inscribed circle: I[3.69992806296; 0.61990472723]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 88° = 1.60657029118 rad
∠ B' = β' = 161° = 0.33216125579 rad
∠ C' = γ' = 111° = 1.20442771839 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 92° ; ; beta = 19° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 92° - 19° = 69° ; ;

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

a = 4.6 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 4.6 * fraction{ sin(19° ) }{ sin (92° ) } = 1.5 ; ;

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 = 4.6 * fraction{ sin(69° ) }{ sin (92° ) } = 4.3 ; ;


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 = 4.6 ; ; b = 1.5 ; ; c = 4.3 ; ;

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

p = a+b+c = 4.6+1.5+4.3 = 10.4 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 10.4 }{ 2 } = 5.2 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 5.2 * (5.2-4.6)(5.2-1.5)(5.2-4.3) } ; ; T = sqrt{ 10.35 } = 3.22 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3.22 }{ 4.6 } = 1.4 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3.22 }{ 1.5 } = 4.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3.22 }{ 4.3 } = 1.5 ; ;

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{ 4.6**2-1.5**2-4.3**2 }{ 2 * 1.5 * 4.3 } ) = 92° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1.5**2-4.6**2-4.3**2 }{ 2 * 4.6 * 4.3 } ) = 19° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 4.3**2-4.6**2-1.5**2 }{ 2 * 1.5 * 4.6 } ) = 69° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3.22 }{ 5.2 } = 0.62 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 4.6 }{ 2 * sin 92° } = 2.3 ; ;




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