Triangle calculator AAS

Please enter two angles and one opposite side
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Right scalene triangle.

Sides: a = 1700   b = 1642.074390469   c = 439.9922376674

Area: T = 361250
Perimeter: p = 3782.066628137
Semiperimeter: s = 1891.033314068

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

Height: ha = 425
Height: hb = 439.9922376674
Height: hc = 1642.074390469

Median: ma = 850
Median: mb = 931.5011459284
Median: mc = 1656.745531276

Inradius: r = 191.0333140683
Circumradius: R = 850

Vertex coordinates: A[439.9922376674; 0] B[0; 0] C[439.9922376674; 1642.074390469]
Centroid: CG[293.3288251116; 547.358796823]
Coordinates of the circumscribed circle: U[219.9966188337; 821.0376952346]
Coordinates of the inscribed circle: I[248.9599235991; 191.0333140683]

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

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

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

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

a = 1700 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 1700 * fraction{ sin(75° ) }{ sin (90° ) } = 1642.07 ; ;

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 = 1700 * fraction{ sin(15° ) }{ sin (90° ) } = 439.99 ; ;


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 = 1700 ; ; b = 1642.07 ; ; c = 439.99 ; ;

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

p = a+b+c = 1700+1642.07+439.99 = 3782.07 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 3782.07 }{ 2 } = 1891.03 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1891.03 * (1891.03-1700)(1891.03-1642.07)(1891.03-439.99) } ; ; T = sqrt{ 130501562500 } = 361250 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 361250 }{ 1700 } = 425 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 361250 }{ 1642.07 } = 439.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 361250 }{ 439.99 } = 1642.07 ; ;

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{ 1700**2-1642.07**2-439.99**2 }{ 2 * 1642.07 * 439.99 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1642.07**2-1700**2-439.99**2 }{ 2 * 1700 * 439.99 } ) = 75° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 439.99**2-1700**2-1642.07**2 }{ 2 * 1642.07 * 1700 } ) = 15° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 361250 }{ 1891.03 } = 191.03 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1700 }{ 2 * sin 90° } = 850 ; ;




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