Triangle calculator ASA

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

Sides: a = 26.97215945477   b = 62.14222654783   c = 75

Area: T = 797.0211495455
Perimeter: p = 164.1143860026
Semiperimeter: s = 82.0576930013

Angle ∠ A = α = 20° = 0.34990658504 rad
Angle ∠ B = β = 52° = 0.9087571211 rad
Angle ∠ C = γ = 108° = 1.88549555922 rad

Height: ha = 59.10108065205
Height: hb = 25.65215107494
Height: hc = 21.25439065455

Median: ma = 67.53986100781
Median: mb = 47.01993382187
Median: mc = 29.80545975583

Inradius: r = 9.71330308839
Circumradius: R = 39.43298334089

Vertex coordinates: A[75; 0] B[0; 0] C[16.60553716911; 21.25439065455]
Centroid: CG[30.5355123897; 7.08546355152]
Coordinates of the circumscribed circle: U[37.5; -12.18444886087]
Coordinates of the inscribed circle: I[19.91546645347; 9.71330308839]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160° = 0.34990658504 rad
∠ B' = β' = 128° = 0.9087571211 rad
∠ C' = γ' = 72° = 1.88549555922 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

alpha = 20° ; ; beta = 52° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 20° - 52° = 108° ; ;

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

c = 75 ; ; ; ; fraction{ a }{ c } = fraction{ sin alpha }{ sin gamma } ; ; ; ; a = c * fraction{ sin alpha }{ sin gamma } ; ; ; ; a = 75 * fraction{ sin 20° }{ sin 108° } = 26.97 ; ;

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 = 75 * fraction{ sin 52° }{ sin 108° } = 62.14 ; ;


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 = 26.97 ; ; b = 62.14 ; ; c = 75 ; ;

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

p = a+b+c = 26.97+62.14+75 = 164.11 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 164.11 }{ 2 } = 82.06 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 82.06 * (82.06-26.97)(82.06-62.14)(82.06-75) } ; ; T = sqrt{ 635243.26 } = 797.02 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 797.02 }{ 26.97 } = 59.1 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 797.02 }{ 62.14 } = 25.65 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 797.02 }{ 75 } = 21.25 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 62.14**2+75**2-26.97**2 }{ 2 * 62.14 * 75 } ) = 20° ; ; b**2 = a**2+c**2 - 2ac cos beta ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 26.97**2+75**2-62.14**2 }{ 2 * 26.97 * 75 } ) = 52° ; ; gamma = 180° - alpha - beta = 180° - 20° - 52° = 108° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 797.02 }{ 82.06 } = 9.71 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin alpha } = fraction{ 26.97 }{ 2 * sin 20° } = 39.43 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 62.14**2+2 * 75**2 - 26.97**2 } }{ 2 } = 67.539 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 75**2+2 * 26.97**2 - 62.14**2 } }{ 2 } = 47.019 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 62.14**2+2 * 26.97**2 - 75**2 } }{ 2 } = 29.805 ; ;
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