Triangle calculator AAS

Please enter two angles and one opposite side
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°


Acute scalene triangle.

Sides: a = 50   b = 43.81550164387   c = 56.67549515915

Area: T = 1052.942242523
Perimeter: p = 150.498996803
Semiperimeter: s = 75.24549840151

Angle ∠ A = α = 58° = 1.01222909662 rad
Angle ∠ B = β = 48° = 0.8387758041 rad
Angle ∠ C = γ = 74° = 1.29215436465 rad

Height: ha = 42.11876970094
Height: hb = 48.06330847969
Height: hc = 37.15772412739

Median: ma = 44.05656795624
Median: mb = 48.74551141405
Median: mc = 37.50882030799

Inradius: r = 13.99435231433
Circumradius: R = 29.47994600841

Vertex coordinates: A[56.67549515915; 0] B[0; 0] C[33.45765303179; 37.15772412739]
Centroid: CG[30.04438273031; 12.38657470913]
Coordinates of the circumscribed circle: U[28.33774757957; 8.12656404285]
Coordinates of the inscribed circle: I[31.43299675764; 13.99435231433]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 122° = 1.01222909662 rad
∠ B' = β' = 132° = 0.8387758041 rad
∠ C' = γ' = 106° = 1.29215436465 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 58° ; ; beta = 48° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 58° - 48° = 74° ; ;

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

a = 50 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 50 * fraction{ sin(48° ) }{ sin (58° ) } = 43.82 ; ;

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 = 50 * fraction{ sin(74° ) }{ sin (58° ) } = 56.67 ; ;


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 = 50 ; ; b = 43.82 ; ; c = 56.67 ; ;

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

p = a+b+c = 50+43.82+56.67 = 150.49 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 150.49 }{ 2 } = 75.24 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 75.24 * (75.24-50)(75.24-43.82)(75.24-56.67) } ; ; T = sqrt{ 1108687.75 } = 1052.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1052.94 }{ 50 } = 42.12 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1052.94 }{ 43.82 } = 48.06 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1052.94 }{ 56.67 } = 37.16 ; ;

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{ 50**2-43.82**2-56.67**2 }{ 2 * 43.82 * 56.67 } ) = 58° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 43.82**2-50**2-56.67**2 }{ 2 * 50 * 56.67 } ) = 48° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 56.67**2-50**2-43.82**2 }{ 2 * 43.82 * 50 } ) = 74° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1052.94 }{ 75.24 } = 13.99 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 50 }{ 2 * sin 58° } = 29.48 ; ;




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