Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 82.45548328038   b = 89.57656632837   c = 35

Area: T = 1442.965957407
Perimeter: p = 207.0330496088
Semiperimeter: s = 103.5155248044

Angle ∠ A = α = 67° = 1.16993705988 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 23° = 0.4011425728 rad

Height: ha = 35
Height: hb = 32.21876698708
Height: hc = 82.45548328038

Median: ma = 54.08804942949
Median: mb = 44.78878316418
Median: mc = 84.29114553956

Inradius: r = 13.94395847601
Circumradius: R = 44.78878316418

Vertex coordinates: A[35; 0] B[0; 0] C[0; 82.45548328038]
Centroid: CG[11.66766666667; 27.48549442679]
Coordinates of the circumscribed circle: U[17.5; 41.22774164019]
Coordinates of the inscribed circle: I[13.94395847601; 13.94395847601]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 113° = 1.16993705988 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 157° = 0.4011425728 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 67° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 67° - 90° = 23° ; ;

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

c = 35 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 35 * fraction{ sin(67° ) }{ sin (23° ) } = 82.45 ; ;

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 = 35 * fraction{ sin(90° ) }{ sin (23° ) } = 89.58 ; ;


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 = 82.45 ; ; b = 89.58 ; ; c = 35 ; ;

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

p = a+b+c = 82.45+89.58+35 = 207.03 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 207.03 }{ 2 } = 103.52 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 103.52 * (103.52-82.45)(103.52-89.58)(103.52-35) } ; ; T = sqrt{ 2082132.33 } = 1442.96 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1442.96 }{ 82.45 } = 35 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1442.96 }{ 89.58 } = 32.22 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1442.96 }{ 35 } = 82.45 ; ;

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{ 82.45**2-89.58**2-35**2 }{ 2 * 89.58 * 35 } ) = 67° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 89.58**2-82.45**2-35**2 }{ 2 * 82.45 * 35 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 35**2-82.45**2-89.58**2 }{ 2 * 89.58 * 82.45 } ) = 23° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1442.96 }{ 103.52 } = 13.94 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 82.45 }{ 2 * sin 67° } = 44.79 ; ;




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