Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 365   b = 142.6176861899   c = 335.984427151

Area: T = 23958.5111225
Perimeter: p = 843.6011133409
Semiperimeter: s = 421.8010566704

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 23° = 0.4011425728 rad
Angle ∠ C = γ = 67° = 1.16993705988 rad

Height: ha = 131.2879513562
Height: hb = 335.984427151
Height: hc = 142.6176861899

Median: ma = 182.5
Median: mb = 343.4688081525
Median: mc = 220.3655439607

Inradius: r = 56.80105667044
Circumradius: R = 182.5

Vertex coordinates: A[335.984427151; 0] B[0; 0] C[335.984427151; 142.6176861899]
Centroid: CG[223.998951434; 47.53989539662]
Coordinates of the circumscribed circle: U[167.9922135755; 71.30884309493]
Coordinates of the inscribed circle: I[279.1843704806; 56.80105667044]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 157° = 0.4011425728 rad
∠ C' = γ' = 113° = 1.16993705988 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 23° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 23° = 67° ; ;

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

a = 365 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 365 * fraction{ sin(23° ) }{ sin (90° ) } = 142.62 ; ;

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 = 365 * fraction{ sin(67° ) }{ sin (90° ) } = 335.98 ; ;


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 = 365 ; ; b = 142.62 ; ; c = 335.98 ; ;

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

p = a+b+c = 365+142.62+335.98 = 843.6 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 843.6 }{ 2 } = 421.8 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 421.8 * (421.8-365)(421.8-142.62)(421.8-335.98) } ; ; T = sqrt{ 574010260.12 } = 23958.51 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 23958.51 }{ 365 } = 131.28 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 23958.51 }{ 142.62 } = 335.98 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 23958.51 }{ 335.98 } = 142.62 ; ;

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{ 365**2-142.62**2-335.98**2 }{ 2 * 142.62 * 335.98 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 142.62**2-365**2-335.98**2 }{ 2 * 365 * 335.98 } ) = 23° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 335.98**2-365**2-142.62**2 }{ 2 * 142.62 * 365 } ) = 67° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 23958.51 }{ 421.8 } = 56.8 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 365 }{ 2 * sin 90° } = 182.5 ; ;




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