Triangle calculator ASA

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

Sides: a = 191.4299343295   b = 62.32332980054   c = 181

Area: T = 5640.258846949
Perimeter: p = 434.75326413
Semiperimeter: s = 217.376632065

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 19° = 0.33216125579 rad
Angle ∠ C = γ = 71° = 1.23991837689 rad

Height: ha = 58.92878359567
Height: hb = 181
Height: hc = 62.32332980054

Median: ma = 95.71546716474
Median: mb = 183.6632866058
Median: mc = 109.884377257

Inradius: r = 25.94769773553
Circumradius: R = 95.71546716474

Vertex coordinates: A[181; 0] B[0; 0] C[181; 62.32332980054]
Centroid: CG[120.6676666667; 20.77444326685]
Coordinates of the circumscribed circle: U[90.5; 31.16216490027]
Coordinates of the inscribed circle: I[155.0533022645; 25.94769773553]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 161° = 0.33216125579 rad
∠ C' = γ' = 109° = 1.23991837689 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 19° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 19° = 71° ; ;

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

c = 181 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 181 * fraction{ sin(90° ) }{ sin (71° ) } = 191.43 ; ;

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 = 181 * fraction{ sin(19° ) }{ sin (71° ) } = 62.32 ; ;


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 = 191.43 ; ; b = 62.32 ; ; c = 181 ; ;

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

p = a+b+c = 191.43+62.32+181 = 434.75 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 434.75 }{ 2 } = 217.38 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 217.38 * (217.38-191.43)(217.38-62.32)(217.38-181) } ; ; T = sqrt{ 31812515.6 } = 5640.26 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5640.26 }{ 191.43 } = 58.93 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5640.26 }{ 62.32 } = 181 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5640.26 }{ 181 } = 62.32 ; ;

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{ 191.43**2-62.32**2-181**2 }{ 2 * 62.32 * 181 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 62.32**2-191.43**2-181**2 }{ 2 * 191.43 * 181 } ) = 19° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 181**2-191.43**2-62.32**2 }{ 2 * 62.32 * 191.43 } ) = 71° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5640.26 }{ 217.38 } = 25.95 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 191.43 }{ 2 * sin 90° } = 95.71 ; ;




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