Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 80   b = 69.97695765712   c = 38.78547696197

Area: T = 1356.877695385
Perimeter: p = 188.7544346191
Semiperimeter: s = 94.37771730954

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 61° = 1.06546508437 rad
Angle ∠ C = γ = 29° = 0.50661454831 rad

Height: ha = 33.92219238463
Height: hb = 38.78547696197
Height: hc = 69.97695765712

Median: ma = 40
Median: mb = 52.23221143152
Median: mc = 72.60772051119

Inradius: r = 14.37771730954
Circumradius: R = 40

Vertex coordinates: A[38.78547696197; 0] B[0; 0] C[38.78547696197; 69.97695765712]
Centroid: CG[25.85765130798; 23.32331921904]
Coordinates of the circumscribed circle: U[19.39223848099; 34.98547882856]
Coordinates of the inscribed circle: I[24.40875965243; 14.37771730954]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 119° = 1.06546508437 rad
∠ C' = γ' = 151° = 0.50661454831 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 61° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 61° = 29° ; ;

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

a = 80 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 80 * fraction{ sin(61° ) }{ sin (90° ) } = 69.97 ; ;

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 = 80 * fraction{ sin(29° ) }{ sin (90° ) } = 38.78 ; ;


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 = 80 ; ; b = 69.97 ; ; c = 38.78 ; ;

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

p = a+b+c = 80+69.97+38.78 = 188.75 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 188.75 }{ 2 } = 94.38 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 94.38 * (94.38-80)(94.38-69.97)(94.38-38.78) } ; ; T = sqrt{ 1841115.07 } = 1356.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1356.88 }{ 80 } = 33.92 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1356.88 }{ 69.97 } = 38.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1356.88 }{ 38.78 } = 69.97 ; ;

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{ 80**2-69.97**2-38.78**2 }{ 2 * 69.97 * 38.78 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 69.97**2-80**2-38.78**2 }{ 2 * 80 * 38.78 } ) = 61° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 38.78**2-80**2-69.97**2 }{ 2 * 69.97 * 80 } ) = 29° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1356.88 }{ 94.38 } = 14.38 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 80 }{ 2 * sin 90° } = 40 ; ;




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