Triangle calculator ASA

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

Sides: a = 46.27880478137   b = 19.43334173385   c = 42

Area: T = 408.1021764108
Perimeter: p = 107.7111465152
Semiperimeter: s = 53.85657325761

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 24.83° = 24°49'48″ = 0.43333652533 rad
Angle ∠ C = γ = 65.17° = 65°10'12″ = 1.13774310735 rad

Height: ha = 17.63769481163
Height: hb = 42
Height: hc = 19.43334173385

Median: ma = 23.13990239069
Median: mb = 43.10993310939
Median: mc = 28.61221951177

Inradius: r = 7.57876847624
Circumradius: R = 23.13990239069

Vertex coordinates: A[42; 0] B[0; 0] C[42; 19.43334173385]
Centroid: CG[28; 6.47878057795]
Coordinates of the circumscribed circle: U[21; 9.71767086692]
Coordinates of the inscribed circle: I[34.42223152376; 7.57876847624]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 155.17° = 155°10'12″ = 0.43333652533 rad
∠ C' = γ' = 114.83° = 114°49'48″ = 1.13774310735 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 24° 49'48" ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 24° 49'48" = 65° 10'12" ; ;

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

c = 42 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 42 * fraction{ sin(90° ) }{ sin (65° 10'12") } = 46.28 ; ;

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 = 42 * fraction{ sin(24° 49'48") }{ sin (65° 10'12") } = 19.43 ; ;


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 = 46.28 ; ; b = 19.43 ; ; c = 42 ; ;

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

p = a+b+c = 46.28+19.43+42 = 107.71 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 107.71 }{ 2 } = 53.86 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 53.86 * (53.86-46.28)(53.86-19.43)(53.86-42) } ; ; T = sqrt{ 166547.05 } = 408.1 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 408.1 }{ 46.28 } = 17.64 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 408.1 }{ 19.43 } = 42 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 408.1 }{ 42 } = 19.43 ; ;

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{ 46.28**2-19.43**2-42**2 }{ 2 * 19.43 * 42 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 19.43**2-46.28**2-42**2 }{ 2 * 46.28 * 42 } ) = 24° 49'48" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 42**2-46.28**2-19.43**2 }{ 2 * 19.43 * 46.28 } ) = 65° 10'12" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 408.1 }{ 53.86 } = 7.58 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 46.28 }{ 2 * sin 90° } = 23.14 ; ;




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