Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 165   b = 1.44399331855   c = 165.0066282934

Area: T = 118.7944487802
Perimeter: p = 331.4466216119
Semiperimeter: s = 165.723310806

Angle ∠ A = α = 89.5° = 89°30' = 1.56220696805 rad
Angle ∠ B = β = 0.5° = 0°30' = 0.00987266463 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 1.44399331855
Height: hb = 165
Height: hc = 1.44398783572

Median: ma = 82.51325651497
Median: mb = 165.0021570756
Median: mc = 82.50331414668

Inradius: r = 0.71768251259
Circumradius: R = 82.50331414668

Vertex coordinates: A[165.0066282934; 0] B[0; 0] C[164.9943717306; 1.44398783572]
Centroid: CG[1100.00000008; 0.48799594524]
Coordinates of the circumscribed circle: U[82.50331414668; 0]
Coordinates of the inscribed circle: I[164.2833174874; 0.71768251259]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90.5° = 90°30' = 1.56220696805 rad
∠ B' = β' = 179.5° = 179°30' = 0.00987266463 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 89° 30' ; ; beta = 0° 30' ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 89° 30' - 0° 30' = 90° ; ;

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

a = 165 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 165 * fraction{ sin(0° 30') }{ sin (89° 30') } = 1.44 ; ;

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 = 165 * fraction{ sin(90° ) }{ sin (89° 30') } = 165.01 ; ;


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 = 165 ; ; b = 1.44 ; ; c = 165.01 ; ;

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

p = a+b+c = 165+1.44+165.01 = 331.45 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 331.45 }{ 2 } = 165.72 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 165.72 * (165.72-165)(165.72-1.44)(165.72-165.01) } ; ; T = sqrt{ 14112.13 } = 118.79 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 118.79 }{ 165 } = 1.44 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 118.79 }{ 1.44 } = 165 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 118.79 }{ 165.01 } = 1.44 ; ;

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{ 165**2-1.44**2-165.01**2 }{ 2 * 1.44 * 165.01 } ) = 89° 30' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1.44**2-165**2-165.01**2 }{ 2 * 165 * 165.01 } ) = 0° 30' ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 165.01**2-165**2-1.44**2 }{ 2 * 1.44 * 165 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 118.79 }{ 165.72 } = 0.72 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 165 }{ 2 * sin 89° 30' } = 82.5 ; ;




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