Triangle calculator AAS

Please enter two angles and one opposite side
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°


Right isosceles triangle.

Sides: a = 342.5   b = 484.3688145113   c = 342.5

Area: T = 58653.125
Perimeter: p = 1169.368814511
Semiperimeter: s = 584.6844072556

Angle ∠ A = α = 45° = 0.78553981634 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 45° = 0.78553981634 rad

Height: ha = 342.5
Height: hb = 242.1844072556
Height: hc = 342.5

Median: ma = 382.9276641147
Median: mb = 242.1844072556
Median: mc = 382.9276641147

Inradius: r = 100.3165927444
Circumradius: R = 242.1844072556

Vertex coordinates: A[342.5; 0] B[0; 0] C[-0; 342.5]
Centroid: CG[114.1676666667; 114.1676666667]
Coordinates of the circumscribed circle: U[171.25; 171.25]
Coordinates of the inscribed circle: I[100.3165927444; 100.3165927444]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 135° = 0.78553981634 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 135° = 0.78553981634 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 45° ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 45° - 90° = 45° ; ;

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

a = 342.5 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 342.5 * fraction{ sin(90° ) }{ sin (45° ) } = 484.37 ; ;

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 = 342.5 * fraction{ sin(45° ) }{ sin (45° ) } = 342.5 ; ;


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 = 342.5 ; ; b = 484.37 ; ; c = 342.5 ; ;

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

p = a+b+c = 342.5+484.37+342.5 = 1169.37 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1169.37 }{ 2 } = 584.68 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 584.68 * (584.68-342.5)(584.68-484.37)(584.68-342.5) } ; ; T = sqrt{ 3440189072.27 } = 58653.13 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 58653.13 }{ 342.5 } = 342.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 58653.13 }{ 484.37 } = 242.18 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 58653.13 }{ 342.5 } = 342.5 ; ;

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{ 342.5**2-484.37**2-342.5**2 }{ 2 * 484.37 * 342.5 } ) = 45° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 484.37**2-342.5**2-342.5**2 }{ 2 * 342.5 * 342.5 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 342.5**2-342.5**2-484.37**2 }{ 2 * 484.37 * 342.5 } ) = 45° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 58653.13 }{ 584.68 } = 100.32 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 342.5 }{ 2 * sin 45° } = 242.18 ; ;




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