Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 18.406625   b = 4.76438880489   c = 17.77990722401

Area: T = 42.34987548828
Perimeter: p = 40.94992102891
Semiperimeter: s = 20.47546051445

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 75° = 1.3098996939 rad

Height: ha = 4.60215625
Height: hb = 17.77990722401
Height: hc = 4.76438880489

Median: ma = 9.2033125
Median: mb = 17.93879225959
Median: mc = 10.08655580794

Inradius: r = 2.06883551445
Circumradius: R = 9.2033125

Vertex coordinates: A[17.77990722401; 0] B[0; 0] C[17.77990722401; 4.76438880489]
Centroid: CG[11.85327148268; 1.5887962683]
Coordinates of the circumscribed circle: U[8.89895361201; 2.38219440245]
Coordinates of the inscribed circle: I[15.71107170956; 2.06883551445]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 105° = 1.3098996939 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 15° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 15° = 75° ; ;

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

a = 18.41 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 18.41 * fraction{ sin(15° ) }{ sin (90° ) } = 4.76 ; ;

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 = 18.41 * fraction{ sin(75° ) }{ sin (90° ) } = 17.78 ; ;


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 = 18.41 ; ; b = 4.76 ; ; c = 17.78 ; ;

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

p = a+b+c = 18.41+4.76+17.78 = 40.95 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 40.95 }{ 2 } = 20.47 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 20.47 * (20.47-18.41)(20.47-4.76)(20.47-17.78) } ; ; T = sqrt{ 1793.42 } = 42.35 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 42.35 }{ 18.41 } = 4.6 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 42.35 }{ 4.76 } = 17.78 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 42.35 }{ 17.78 } = 4.76 ; ;

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{ 18.41**2-4.76**2-17.78**2 }{ 2 * 4.76 * 17.78 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 4.76**2-18.41**2-17.78**2 }{ 2 * 18.41 * 17.78 } ) = 15° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 17.78**2-18.41**2-4.76**2 }{ 2 * 4.76 * 18.41 } ) = 75° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 42.35 }{ 20.47 } = 2.07 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 18.41 }{ 2 * sin 90° } = 9.2 ; ;




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