Triangle calculator ASA

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

Sides: a = 420.1254522926   b = 732.4654753257   c = 600

Area: T = 126037.3576878
Perimeter: p = 1752.589927618
Semiperimeter: s = 876.2954638091

Angle ∠ A = α = 35° = 0.61108652382 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 55° = 0.96599310886 rad

Height: ha = 600
Height: hb = 344.1465861811
Height: hc = 420.1254522926

Median: ma = 635.7099173829
Median: mb = 366.2322376628
Median: mc = 516.2410849569

Inradius: r = 143.8329884835
Circumradius: R = 366.2322376628

Vertex coordinates: A[600; 0] B[0; 0] C[-0; 420.1254522926]
Centroid: CG[200; 140.0421507642]
Coordinates of the circumscribed circle: U[300; 210.0622261463]
Coordinates of the inscribed circle: I[143.8329884835; 143.8329884835]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145° = 0.61108652382 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 125° = 0.96599310886 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 35° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 35° - 90° = 55° ; ;

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

c = 600 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 600 * fraction{ sin(35° ) }{ sin (55° ) } = 420.12 ; ;

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 = 600 * fraction{ sin(90° ) }{ sin (55° ) } = 732.46 ; ;


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 = 420.12 ; ; b = 732.46 ; ; c = 600 ; ;

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

p = a+b+c = 420.12+732.46+600 = 1752.59 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1752.59 }{ 2 } = 876.29 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 876.29 * (876.29-420.12)(876.29-732.46)(876.29-600) } ; ; T = sqrt{ 15885415328.7 } = 126037.36 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 126037.36 }{ 420.12 } = 600 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 126037.36 }{ 732.46 } = 344.15 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 126037.36 }{ 600 } = 420.12 ; ;

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{ 420.12**2-732.46**2-600**2 }{ 2 * 732.46 * 600 } ) = 35° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 732.46**2-420.12**2-600**2 }{ 2 * 420.12 * 600 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 600**2-420.12**2-732.46**2 }{ 2 * 732.46 * 420.12 } ) = 55° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 126037.36 }{ 876.29 } = 143.83 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 420.12 }{ 2 * sin 35° } = 366.23 ; ;




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