Triangle calculator ASA

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

Sides: a = 2581.268813827   b = 3164.252157518   c = 800

Area: T = 779243.117723
Perimeter: p = 6545.521971345
Semiperimeter: s = 3272.765985672

Angle ∠ A = α = 38° = 0.66332251158 rad
Angle ∠ B = β = 131° = 2.28663813201 rad
Angle ∠ C = γ = 11° = 0.19219862177 rad

Height: ha = 603.7687664178
Height: hb = 492.5299180261
Height: hc = 1948.108779308

Median: ma = 1913.245533584
Median: mb = 1071.611121358
Median: mc = 2859.671071816

Inradius: r = 238.1099693025
Circumradius: R = 2096.337722567

Vertex coordinates: A[800; 0] B[0; 0] C[-1693.464426836; 1948.108779308]
Centroid: CG[-297.8211422786; 649.3699264358]
Coordinates of the circumscribed circle: U[400; 2057.822160639]
Coordinates of the inscribed circle: I[108.5088281546; 238.1099693025]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 142° = 0.66332251158 rad
∠ B' = β' = 49° = 2.28663813201 rad
∠ C' = γ' = 169° = 0.19219862177 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 38° ; ; beta = 131° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 38° - 131° = 11° ; ;

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

c = 800 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 800 * fraction{ sin(38° ) }{ sin (11° ) } = 2581.27 ; ;

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 = 800 * fraction{ sin(131° ) }{ sin (11° ) } = 3164.25 ; ;


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 = 2581.27 ; ; b = 3164.25 ; ; c = 800 ; ;

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

p = a+b+c = 2581.27+3164.25+800 = 6545.52 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 6545.52 }{ 2 } = 3272.76 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3272.76 * (3272.76-2581.27)(3272.76-3164.25)(3272.76-800) } ; ; T = sqrt{ 607219835750 } = 779243.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 779243.12 }{ 2581.27 } = 603.77 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 779243.12 }{ 3164.25 } = 492.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 779243.12 }{ 800 } = 1948.11 ; ;

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{ 2581.27**2-3164.25**2-800**2 }{ 2 * 3164.25 * 800 } ) = 38° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 3164.25**2-2581.27**2-800**2 }{ 2 * 2581.27 * 800 } ) = 131° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 800**2-2581.27**2-3164.25**2 }{ 2 * 3164.25 * 2581.27 } ) = 11° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 779243.12 }{ 3272.76 } = 238.1 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2581.27 }{ 2 * sin 38° } = 2096.34 ; ;




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