Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 2.1   b = 1.26438115486   c = 1.67771345711

Area: T = 1.06597910198
Perimeter: p = 5.04109461197
Semiperimeter: s = 2.52204730599

Angle ∠ A = α = 90° = 1.57107963268 rad
Angle ∠ B = β = 37° = 0.64657718232 rad
Angle ∠ C = γ = 53° = 0.92550245036 rad

Height: ha = 1.00993247807
Height: hb = 1.67771345711
Height: hc = 1.26438115486

Median: ma = 1.05
Median: mb = 1.79222291364
Median: mc = 1.51767118127

Inradius: r = 0.42204730599
Circumradius: R = 1.05

Vertex coordinates: A[1.67771345711; 0] B[0; 0] C[1.67771345711; 1.26438115486]
Centroid: CG[1.11880897141; 0.42112705162]
Coordinates of the circumscribed circle: U[0.83985672855; 0.63219057743]
Coordinates of the inscribed circle: I[1.25766615112; 0.42204730599]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 90° = 1.57107963268 rad
∠ B' = β' = 143° = 0.64657718232 rad
∠ C' = γ' = 127° = 0.92550245036 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 90° ; ; beta = 37° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 90° - 37° = 53° ; ;

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

a = 2.1 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 2.1 * fraction{ sin(37° ) }{ sin (90° ) } = 1.26 ; ;

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 = 2.1 * fraction{ sin(53° ) }{ sin (90° ) } = 1.68 ; ;


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 = 2.1 ; ; b = 1.26 ; ; c = 1.68 ; ;

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

p = a+b+c = 2.1+1.26+1.68 = 5.04 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 5.04 }{ 2 } = 2.52 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 2.52 * (2.52-2.1)(2.52-1.26)(2.52-1.68) } ; ; T = sqrt{ 1.12 } = 1.06 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1.06 }{ 2.1 } = 1.01 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1.06 }{ 1.26 } = 1.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1.06 }{ 1.68 } = 1.26 ; ;

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{ 2.1**2-1.26**2-1.68**2 }{ 2 * 1.26 * 1.68 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1.26**2-2.1**2-1.68**2 }{ 2 * 2.1 * 1.68 } ) = 37° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1.68**2-2.1**2-1.26**2 }{ 2 * 1.26 * 2.1 } ) = 53° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1.06 }{ 2.52 } = 0.42 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 2.1 }{ 2 * sin 90° } = 1.05 ; ;




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