Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 11.25   b = 12.86327332636   c = 6.23659768288

Area: T = 35.07773696622
Perimeter: p = 30.34987100924
Semiperimeter: s = 15.17443550462

Angle ∠ A = α = 61° = 1.06546508437 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 29° = 0.50661454831 rad

Height: ha = 6.23659768288
Height: hb = 5.45441082278
Height: hc = 11.25

Median: ma = 8.39880969279
Median: mb = 6.43113666318
Median: mc = 11.67440889046

Inradius: r = 2.31216217826
Circumradius: R = 6.43113666318

Vertex coordinates: A[6.23659768288; 0] B[0; 0] C[-0; 11.25]
Centroid: CG[2.07986589429; 3.75]
Coordinates of the circumscribed circle: U[3.11879884144; 5.625]
Coordinates of the inscribed circle: I[2.31216217826; 2.31216217826]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 119° = 1.06546508437 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 151° = 0.50661454831 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 61° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 61° - 90° = 29° ; ;

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

a = 11.25 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 11.25 * fraction{ sin(90° ) }{ sin (61° ) } = 12.86 ; ;

3. By using the law of sines, we calculate last unknown side c

 fraction{ c }{ a } = fraction{ sin( gamma ) }{ sin ( alpha ) } ; ; ; ; c = a * fraction{ sin( gamma ) }{ sin ( alpha ) } ; ; ; ; c = 11.25 * fraction{ sin(29° ) }{ sin (61° ) } = 6.24 ; ;


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 = 11.25 ; ; b = 12.86 ; ; c = 6.24 ; ;

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

p = a+b+c = 11.25+12.86+6.24 = 30.35 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 30.35 }{ 2 } = 15.17 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 15.17 * (15.17-11.25)(15.17-12.86)(15.17-6.24) } ; ; T = sqrt{ 1230.42 } = 35.08 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 35.08 }{ 11.25 } = 6.24 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 35.08 }{ 12.86 } = 5.45 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 35.08 }{ 6.24 } = 11.25 ; ;

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{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 11.25**2-12.86**2-6.24**2 }{ 2 * 12.86 * 6.24 } ) = 61° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 12.86**2-11.25**2-6.24**2 }{ 2 * 11.25 * 6.24 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 6.24**2-11.25**2-12.86**2 }{ 2 * 12.86 * 11.25 } ) = 29° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 35.08 }{ 15.17 } = 2.31 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 11.25 }{ 2 * sin 61° } = 6.43 ; ;




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