Triangle calculator ASA

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

Sides: a = 51.18220037316   b = 24.33551976962   c = 65

Area: T = 568.9211478189
Perimeter: p = 140.5177201428
Semiperimeter: s = 70.25986007139

Angle ∠ A = α = 46° = 0.80328514559 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 114° = 1.99896753473 rad

Height: ha = 22.23113093162
Height: hb = 46.7577087022
Height: hc = 17.5055276252

Median: ma = 41.87772199049
Median: mb = 57.22110476246
Median: mc = 23.44546086861

Inradius: r = 8.0987534998
Circumradius: R = 35.57656790514

Vertex coordinates: A[65; 0] B[0; 0] C[48.09553512236; 17.5055276252]
Centroid: CG[37.69884504079; 5.8355092084]
Coordinates of the circumscribed circle: U[32.5; -14.47699322725]
Coordinates of the inscribed circle: I[45.92334030177; 8.0987534998]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 134° = 0.80328514559 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 66° = 1.99896753473 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 46° ; ; beta = 20° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 46° - 20° = 114° ; ;

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

c = 65 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 65 * fraction{ sin(46° ) }{ sin (114° ) } = 51.18 ; ;

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 = 65 * fraction{ sin(20° ) }{ sin (114° ) } = 24.34 ; ;


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 = 51.18 ; ; b = 24.34 ; ; c = 65 ; ;

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

p = a+b+c = 51.18+24.34+65 = 140.52 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 140.52 }{ 2 } = 70.26 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 70.26 * (70.26-51.18)(70.26-24.34)(70.26-65) } ; ; T = sqrt{ 323671.65 } = 568.92 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 568.92 }{ 51.18 } = 22.23 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 568.92 }{ 24.34 } = 46.76 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 568.92 }{ 65 } = 17.51 ; ;

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{ 51.18**2-24.34**2-65**2 }{ 2 * 24.34 * 65 } ) = 46° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24.34**2-51.18**2-65**2 }{ 2 * 51.18 * 65 } ) = 20° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 65**2-51.18**2-24.34**2 }{ 2 * 24.34 * 51.18 } ) = 114° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 568.92 }{ 70.26 } = 8.1 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 51.18 }{ 2 * sin 46° } = 35.58 ; ;




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