Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Acute scalene triangle.

Sides: a = 90   b = 79.61096737803   c = 97.65657217619

Area: T = 3366.388803475
Perimeter: p = 267.2655395542
Semiperimeter: s = 133.6332697771

Angle ∠ A = α = 60° = 1.04771975512 rad
Angle ∠ B = β = 50° = 0.8732664626 rad
Angle ∠ C = γ = 70° = 1.22217304764 rad

Height: ha = 74.80986229945
Height: hb = 84.57223358707
Height: hc = 68.94439998807

Median: ma = 76.89106371161
Median: mb = 85.05223071796
Median: mc = 69.5321935695

Inradius: r = 25.19113498036
Circumradius: R = 51.96215242271

Vertex coordinates: A[97.65657217619; 0] B[0; 0] C[57.85108848718; 68.94439998807]
Centroid: CG[51.83655355446; 22.98113332936]
Coordinates of the circumscribed circle: U[48.8287860881; 17.77218879636]
Coordinates of the inscribed circle: I[54.02330239908; 25.19113498036]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 120° = 1.04771975512 rad
∠ B' = β' = 130° = 0.8732664626 rad
∠ C' = γ' = 110° = 1.22217304764 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 60° ; ; beta = 50° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 60° - 50° = 70° ; ;

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

a = 90 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 90 * fraction{ sin(50° ) }{ sin (60° ) } = 79.61 ; ;

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 = 90 * fraction{ sin(70° ) }{ sin (60° ) } = 97.66 ; ;


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 = 90 ; ; b = 79.61 ; ; c = 97.66 ; ;

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

p = a+b+c = 90+79.61+97.66 = 267.27 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 267.27 }{ 2 } = 133.63 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 133.63 * (133.63-90)(133.63-79.61)(133.63-97.66) } ; ; T = sqrt{ 11332568.4 } = 3366.39 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 3366.39 }{ 90 } = 74.81 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 3366.39 }{ 79.61 } = 84.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 3366.39 }{ 97.66 } = 68.94 ; ;

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{ 90**2-79.61**2-97.66**2 }{ 2 * 79.61 * 97.66 } ) = 60° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 79.61**2-90**2-97.66**2 }{ 2 * 90 * 97.66 } ) = 50° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 97.66**2-90**2-79.61**2 }{ 2 * 79.61 * 90 } ) = 70° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 3366.39 }{ 133.63 } = 25.19 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 90 }{ 2 * sin 60° } = 51.96 ; ;




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