Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 123.3   b = 130.3811141511   c = 42.38333937014

Area: T = 2612.936622169
Perimeter: p = 296.0654535212
Semiperimeter: s = 148.0322267606

Angle ∠ A = α = 71.03° = 71°1'48″ = 1.24397073677 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 18.97° = 18°58'12″ = 0.33110889591 rad

Height: ha = 42.38333937014
Height: hb = 40.08215055217
Height: hc = 123.3

Median: ma = 74.81435987749
Median: mb = 65.19105707554
Median: mc = 125.1087865522

Inradius: r = 17.65111260953
Circumradius: R = 65.19105707554

Vertex coordinates: A[42.38333937014; 0] B[0; 0] C[-0; 123.3]
Centroid: CG[14.12877979005; 41.1]
Coordinates of the circumscribed circle: U[21.19216968507; 61.65]
Coordinates of the inscribed circle: I[17.65111260953; 17.65111260953]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108.97° = 108°58'12″ = 1.24397073677 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 161.03° = 161°1'48″ = 0.33110889591 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 71° 1'48" ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 71° 1'48" - 90° = 18° 58'12" ; ;

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

a = 123.3 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 123.3 * fraction{ sin(90° ) }{ sin (71° 1'48") } = 130.38 ; ;

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 = 123.3 * fraction{ sin(18° 58'12") }{ sin (71° 1'48") } = 42.38 ; ;


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 = 123.3 ; ; b = 130.38 ; ; c = 42.38 ; ;

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

p = a+b+c = 123.3+130.38+42.38 = 296.06 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 296.06 }{ 2 } = 148.03 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 148.03 * (148.03-123.3)(148.03-130.38)(148.03-42.38) } ; ; T = sqrt{ 6827435.7 } = 2612.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2612.94 }{ 123.3 } = 42.38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2612.94 }{ 130.38 } = 40.08 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2612.94 }{ 42.38 } = 123.3 ; ;

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{ 123.3**2-130.38**2-42.38**2 }{ 2 * 130.38 * 42.38 } ) = 71° 1'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 130.38**2-123.3**2-42.38**2 }{ 2 * 123.3 * 42.38 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 42.38**2-123.3**2-130.38**2 }{ 2 * 130.38 * 123.3 } ) = 18° 58'12" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2612.94 }{ 148.03 } = 17.65 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 123.3 }{ 2 * sin 71° 1'48" } = 65.19 ; ;




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