Triangle calculator AAS

Please enter two angles and one opposite side
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Acute isosceles triangle.

Sides: a = 360   b = 360   c = 222.492223595

Area: T = 38088.48443486
Perimeter: p = 942.492223595
Semiperimeter: s = 471.2466117975

Angle ∠ A = α = 72° = 1.25766370614 rad
Angle ∠ B = β = 72° = 1.25766370614 rad
Angle ∠ C = γ = 36° = 0.62883185307 rad

Height: ha = 211.6032690825
Height: hb = 211.6032690825
Height: hc = 342.3880345866

Median: ma = 239.0643584699
Median: mb = 239.0643584699
Median: mc = 342.3880345866

Inradius: r = 80.82550357843
Circumradius: R = 189.2633200363

Vertex coordinates: A[222.492223595; 0] B[0; 0] C[111.2466117975; 342.3880345866]
Centroid: CG[111.2466117975; 114.1276781955]
Coordinates of the circumscribed circle: U[111.2466117975; 153.1177145503]
Coordinates of the inscribed circle: I[111.2466117975; 80.82550357843]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108° = 1.25766370614 rad
∠ B' = β' = 108° = 1.25766370614 rad
∠ C' = γ' = 144° = 0.62883185307 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 72° ; ; beta = 72° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 72° - 72° = 36° ; ;

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

a = 360 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 360 * fraction{ sin(72° ) }{ sin (72° ) } = 360 ; ;

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 = 360 * fraction{ sin(36° ) }{ sin (72° ) } = 222.49 ; ;


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 = 360 ; ; b = 360 ; ; c = 222.49 ; ;

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

p = a+b+c = 360+360+222.49 = 942.49 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 942.49 }{ 2 } = 471.25 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 471.25 * (471.25-360)(471.25-360)(471.25-222.49) } ; ; T = sqrt{ 1450732639.97 } = 38088.48 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 38088.48 }{ 360 } = 211.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 38088.48 }{ 360 } = 211.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 38088.48 }{ 222.49 } = 342.38 ; ;

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{ 360**2-360**2-222.49**2 }{ 2 * 360 * 222.49 } ) = 72° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 360**2-360**2-222.49**2 }{ 2 * 360 * 222.49 } ) = 72° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 222.49**2-360**2-360**2 }{ 2 * 360 * 360 } ) = 36° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 38088.48 }{ 471.25 } = 80.83 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 360 }{ 2 * sin 72° } = 189.26 ; ;




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