Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
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Obtuse scalene triangle.

Sides: a = 89.64659525893   b = 75.08435076026   c = 158

Area: T = 1832.964430787
Perimeter: p = 322.7299460192
Semiperimeter: s = 161.3654730096

Angle ∠ A = α = 18° = 0.31441592654 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 147° = 2.56656340004 rad

Height: ha = 40.89334091262
Height: hb = 48.82546851112
Height: hc = 23.20220798465

Median: ma = 115.289949368
Median: mb = 122.8454678881
Median: mc = 24.41223936715

Inradius: r = 11.35991384361
Circumradius: R = 145.0550198243

Vertex coordinates: A[158; 0] B[0; 0] C[86.59113408283; 23.20220798465]
Centroid: CG[81.53304469428; 7.73440266155]
Coordinates of the circumscribed circle: U[79; -121.6499332141]
Coordinates of the inscribed circle: I[86.28112224934; 11.35991384361]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162° = 0.31441592654 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 33° = 2.56656340004 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 18° ; ; beta = 15° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 18° - 15° = 147° ; ;

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

c = 158 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 158 * fraction{ sin(18° ) }{ sin (147° ) } = 89.65 ; ;

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

 fraction{ b }{ c } = fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = c * fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = 158 * fraction{ sin(15° ) }{ sin (147° ) } = 75.08 ; ;


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 = 89.65 ; ; b = 75.08 ; ; c = 158 ; ;

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

p = a+b+c = 89.65+75.08+158 = 322.73 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 322.73 }{ 2 } = 161.36 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 161.36 * (161.36-89.65)(161.36-75.08)(161.36-158) } ; ; T = sqrt{ 3359758.15 } = 1832.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1832.96 }{ 89.65 } = 40.89 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1832.96 }{ 75.08 } = 48.82 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1832.96 }{ 158 } = 23.2 ; ;

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{ 89.65**2-75.08**2-158**2 }{ 2 * 75.08 * 158 } ) = 18° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 75.08**2-89.65**2-158**2 }{ 2 * 89.65 * 158 } ) = 15° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 158**2-89.65**2-75.08**2 }{ 2 * 75.08 * 89.65 } ) = 147° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1832.96 }{ 161.36 } = 11.36 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 89.65 }{ 2 * sin 18° } = 145.05 ; ;




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