Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
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°


Right scalene triangle.

Sides: a = 97.47659088699   b = 315.4398667271   c = 300

Area: T = 14621.38663305
Perimeter: p = 712.9154576141
Semiperimeter: s = 356.4577288071

Angle ∠ A = α = 18° = 0.31441592654 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 72° = 1.25766370614 rad

Height: ha = 300
Height: hb = 92.70550983125
Height: hc = 97.47659088699

Median: ma = 303.9333196941
Median: mb = 157.7199333636
Median: mc = 178.8989778383

Inradius: r = 41.01986207992
Circumradius: R = 157.7199333636

Vertex coordinates: A[300; 0] B[0; 0] C[-0; 97.47659088699]
Centroid: CG[100; 32.49219696233]
Coordinates of the circumscribed circle: U[150; 48.73879544349]
Coordinates of the inscribed circle: I[41.01986207992; 41.01986207992]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162° = 0.31441592654 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 108° = 1.25766370614 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 18° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 18° - 90° = 72° ; ;

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

c = 300 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 300 * fraction{ sin(18° ) }{ sin (72° ) } = 97.48 ; ;

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 = 300 * fraction{ sin(90° ) }{ sin (72° ) } = 315.44 ; ;


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 = 97.48 ; ; b = 315.44 ; ; c = 300 ; ;

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

p = a+b+c = 97.48+315.44+300 = 712.91 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 712.91 }{ 2 } = 356.46 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 356.46 * (356.46-97.48)(356.46-315.44)(356.46-300) } ; ; T = sqrt{ 213784938.23 } = 14621.39 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 14621.39 }{ 97.48 } = 300 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 14621.39 }{ 315.44 } = 92.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 14621.39 }{ 300 } = 97.48 ; ;

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{ 97.48**2-315.44**2-300**2 }{ 2 * 315.44 * 300 } ) = 18° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 315.44**2-97.48**2-300**2 }{ 2 * 97.48 * 300 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 300**2-97.48**2-315.44**2 }{ 2 * 315.44 * 97.48 } ) = 72° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 14621.39 }{ 356.46 } = 41.02 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 97.48 }{ 2 * sin 18° } = 157.72 ; ;




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