Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 34   b = 74.89114349959   c = 66.72987571872

Area: T = 1134.389887218
Perimeter: p = 175.6220192183
Semiperimeter: s = 87.81100960915

Angle ∠ A = α = 27° = 0.4711238898 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 63° = 1.10995574288 rad

Height: ha = 66.72987571872
Height: hb = 30.29442218224
Height: hc = 34

Median: ma = 68.86601992137
Median: mb = 37.44657174979
Median: mc = 47.6365929286

Inradius: r = 12.91986610956
Circumradius: R = 37.44657174979

Vertex coordinates: A[66.72987571872; 0] B[0; 0] C[-0; 34]
Centroid: CG[22.24329190624; 11.33333333333]
Coordinates of the circumscribed circle: U[33.36443785936; 17]
Coordinates of the inscribed circle: I[12.91986610956; 12.91986610956]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 153° = 0.4711238898 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 117° = 1.10995574288 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 27° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 27° - 90° = 63° ; ;

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

a = 34 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 34 * fraction{ sin(90° ) }{ sin (27° ) } = 74.89 ; ;

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 = 34 * fraction{ sin(63° ) }{ sin (27° ) } = 66.73 ; ;


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 = 34 ; ; b = 74.89 ; ; c = 66.73 ; ;

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

p = a+b+c = 34+74.89+66.73 = 175.62 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 175.62 }{ 2 } = 87.81 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 87.81 * (87.81-34)(87.81-74.89)(87.81-66.73) } ; ; T = sqrt{ 1286838.11 } = 1134.39 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1134.39 }{ 34 } = 66.73 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1134.39 }{ 74.89 } = 30.29 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1134.39 }{ 66.73 } = 34 ; ;

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{ 34**2-74.89**2-66.73**2 }{ 2 * 74.89 * 66.73 } ) = 27° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 74.89**2-34**2-66.73**2 }{ 2 * 34 * 66.73 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 66.73**2-34**2-74.89**2 }{ 2 * 74.89 * 34 } ) = 63° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1134.39 }{ 87.81 } = 12.92 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 34 }{ 2 * sin 27° } = 37.45 ; ;




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