Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Acute scalene triangle.

Sides: a = 68   b = 57.66554474838   c = 81.99106482914

Area: T = 1936.487666459
Perimeter: p = 207.6566095775
Semiperimeter: s = 103.8288047888

Angle ∠ A = α = 55° = 0.96599310886 rad
Angle ∠ B = β = 44° = 0.76879448709 rad
Angle ∠ C = γ = 81° = 1.41437166941 rad

Height: ha = 56.9555490135
Height: hb = 67.16328071605
Height: hc = 47.23767691912

Median: ma = 62.19223236451
Median: mb = 69.5843814535
Median: mc = 47.89660887228

Inradius: r = 18.65109012159
Circumradius: R = 41.50663360179

Vertex coordinates: A[81.99106482914; 0] B[0; 0] C[48.9155106423; 47.23767691912]
Centroid: CG[43.63552515715; 15.74655897304]
Coordinates of the circumscribed circle: U[40.99553241457; 6.49330214707]
Coordinates of the inscribed circle: I[46.16326004038; 18.65109012159]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 125° = 0.96599310886 rad
∠ B' = β' = 136° = 0.76879448709 rad
∠ C' = γ' = 99° = 1.41437166941 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 55° ; ; beta = 44° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 55° - 44° = 81° ; ;

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

a = 68 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 68 * fraction{ sin(44° ) }{ sin (55° ) } = 57.67 ; ;

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 = 68 * fraction{ sin(81° ) }{ sin (55° ) } = 81.99 ; ;


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 = 68 ; ; b = 57.67 ; ; c = 81.99 ; ;

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

p = a+b+c = 68+57.67+81.99 = 207.66 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 207.66 }{ 2 } = 103.83 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 103.83 * (103.83-68)(103.83-57.67)(103.83-81.99) } ; ; T = sqrt{ 3749980.6 } = 1936.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1936.49 }{ 68 } = 56.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1936.49 }{ 57.67 } = 67.16 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1936.49 }{ 81.99 } = 47.24 ; ;

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{ 68**2-57.67**2-81.99**2 }{ 2 * 57.67 * 81.99 } ) = 55° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 57.67**2-68**2-81.99**2 }{ 2 * 68 * 81.99 } ) = 44° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 81.99**2-68**2-57.67**2 }{ 2 * 57.67 * 68 } ) = 81° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1936.49 }{ 103.83 } = 18.65 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 68 }{ 2 * sin 55° } = 41.51 ; ;




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