Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 3061.467   b = 2828.427670076   c = 1171.573269963

Area: T = 1656853.753276
Perimeter: p = 7061.466640039
Semiperimeter: s = 3530.73332002

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 67.5° = 67°30' = 1.17880972451 rad
Angle ∠ C = γ = 22.5° = 22°30' = 0.39326990817 rad

Height: ha = 1082.392203804
Height: hb = 1171.573269963
Height: hc = 2828.427670076

Median: ma = 1530.73435
Median: mb = 1836.459909046
Median: mc = 2888.454997346

Inradius: r = 469.2666200196
Circumradius: R = 1530.73435

Vertex coordinates: A[1171.573269963; 0] B[0; 0] C[1171.573269963; 2828.427670076]
Centroid: CG[781.0488466422; 942.8098900253]
Coordinates of the circumscribed circle: U[585.7866349816; 1414.213335038]
Coordinates of the inscribed circle: I[702.3066499437; 469.2666200196]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 112.5° = 112°30' = 1.17880972451 rad
∠ C' = γ' = 157.5° = 157°30' = 0.39326990817 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 67° 30' ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 67° 30' = 22° 30' ; ;

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

a = 3061.47 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 3061.47 * fraction{ sin(67° 30') }{ sin (90° ) } = 2828.43 ; ;

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 = 3061.47 * fraction{ sin(22° 30') }{ sin (90° ) } = 1171.57 ; ;


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 = 3061.47 ; ; b = 2828.43 ; ; c = 1171.57 ; ;

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

p = a+b+c = 3061.47+2828.43+1171.57 = 7061.47 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 7061.47 }{ 2 } = 3530.73 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3530.73 * (3530.73-3061.47)(3530.73-2828.43)(3530.73-1171.57) } ; ; T = sqrt{ 2.745 * 10**{ 12 } } = 1656853.75 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1656853.75 }{ 3061.47 } = 1082.39 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1656853.75 }{ 2828.43 } = 1171.57 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1656853.75 }{ 1171.57 } = 2828.43 ; ;

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{ 3061.47**2-2828.43**2-1171.57**2 }{ 2 * 2828.43 * 1171.57 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 2828.43**2-3061.47**2-1171.57**2 }{ 2 * 3061.47 * 1171.57 } ) = 67° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1171.57**2-3061.47**2-2828.43**2 }{ 2 * 2828.43 * 3061.47 } ) = 22° 30' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1656853.75 }{ 3530.73 } = 469.27 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3061.47 }{ 2 * sin 90° } = 1530.73 ; ;




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