Triangle calculator ASA

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


Right scalene triangle.

Sides: a = 5.78664666378   b = 1.97990881488   c = 5.43875

Area: T = 5.38106459046
Perimeter: p = 13.20330547867
Semiperimeter: s = 6.60215273933

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 20° = 0.34990658504 rad
Angle ∠ C = γ = 70° = 1.22217304764 rad

Height: ha = 1.86597345293
Height: hb = 5.43875
Height: hc = 1.97990881488

Median: ma = 2.89332333189
Median: mb = 5.52768077337
Median: mc = 3.3632795186

Inradius: r = 0.81550607555
Circumradius: R = 2.89332333189

Vertex coordinates: A[5.43875; 0] B[0; 0] C[5.43875; 1.97990881488]
Centroid: CG[3.625; 0.66596960496]
Coordinates of the circumscribed circle: U[2.719875; 0.99895440744]
Coordinates of the inscribed circle: I[4.62224392445; 0.81550607555]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 160° = 0.34990658504 rad
∠ C' = γ' = 110° = 1.22217304764 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 20° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 20° = 70° ; ;

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

c = 5.44 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 5.44 * fraction{ sin(90° ) }{ sin (70° ) } = 5.79 ; ;

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 = 5.44 * fraction{ sin(20° ) }{ sin (70° ) } = 1.98 ; ;


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 = 5.79 ; ; b = 1.98 ; ; c = 5.44 ; ;

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

p = a+b+c = 5.79+1.98+5.44 = 13.2 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 13.2 }{ 2 } = 6.6 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 6.6 * (6.6-5.79)(6.6-1.98)(6.6-5.44) } ; ; T = sqrt{ 28.95 } = 5.38 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 5.38 }{ 5.79 } = 1.86 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 5.38 }{ 1.98 } = 5.44 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 5.38 }{ 5.44 } = 1.98 ; ;

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{ 5.79**2-1.98**2-5.44**2 }{ 2 * 1.98 * 5.44 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1.98**2-5.79**2-5.44**2 }{ 2 * 5.79 * 5.44 } ) = 20° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 5.44**2-5.79**2-1.98**2 }{ 2 * 1.98 * 5.79 } ) = 70° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 5.38 }{ 6.6 } = 0.82 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 5.79 }{ 2 * sin 90° } = 2.89 ; ;




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