Triangle calculator ASA

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

Sides: a = 58.01215766531   b = 334.075535531   c = 329

Area: T = 9542.904435943
Perimeter: p = 721.0876931963
Semiperimeter: s = 360.5433465982

Angle ∠ A = α = 10° = 0.17545329252 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 80° = 1.39662634016 rad

Height: ha = 329
Height: hb = 57.13302504524
Height: hc = 58.01215766531

Median: ma = 330.2766150753
Median: mb = 167.0387677655
Median: mc = 174.4299335336

Inradius: r = 26.46881106713
Circumradius: R = 167.0387677655

Vertex coordinates: A[329; 0] B[0; 0] C[-0; 58.01215766531]
Centroid: CG[109.6676666667; 19.33771922177]
Coordinates of the circumscribed circle: U[164.5; 29.00657883265]
Coordinates of the inscribed circle: I[26.46881106713; 26.46881106713]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 170° = 0.17545329252 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 100° = 1.39662634016 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 10° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 10° - 90° = 80° ; ;

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

c = 329 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 329 * fraction{ sin(10° ) }{ sin (80° ) } = 58.01 ; ;

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 = 329 * fraction{ sin(90° ) }{ sin (80° ) } = 334.08 ; ;


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 = 58.01 ; ; b = 334.08 ; ; c = 329 ; ;

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

p = a+b+c = 58.01+334.08+329 = 721.09 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 721.09 }{ 2 } = 360.54 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 360.54 * (360.54-58.01)(360.54-334.08)(360.54-329) } ; ; T = sqrt{ 91067023.61 } = 9542.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 9542.9 }{ 58.01 } = 329 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 9542.9 }{ 334.08 } = 57.13 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 9542.9 }{ 329 } = 58.01 ; ;

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{ 58.01**2-334.08**2-329**2 }{ 2 * 334.08 * 329 } ) = 10° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 334.08**2-58.01**2-329**2 }{ 2 * 58.01 * 329 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 329**2-58.01**2-334.08**2 }{ 2 * 334.08 * 58.01 } ) = 80° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 9542.9 }{ 360.54 } = 26.47 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 58.01 }{ 2 * sin 10° } = 167.04 ; ;




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