Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 1.125   b = 0.89884649488   c = 0.6777041901

Area: T = 0.30441492085
Perimeter: p = 2.70105068498
Semiperimeter: s = 1.35502534249

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

Height: ha = 0.5410709704
Height: hb = 0.6777041901
Height: hc = 0.89884649488

Median: ma = 0.56325
Median: mb = 0.81325241854
Median: mc = 0.96601227516

Inradius: r = 0.22552534249
Circumradius: R = 0.56325

Vertex coordinates: A[0.6777041901; 0] B[0; 0] C[0.6777041901; 0.89884649488]
Centroid: CG[0.45113612674; 0.29994883163]
Coordinates of the circumscribed circle: U[0.33985209505; 0.44992324744]
Coordinates of the inscribed circle: I[0.45217884761; 0.22552534249]

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

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

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

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

a = 1.13 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 1.13 * fraction{ sin(53° ) }{ sin (90° ) } = 0.9 ; ;

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 = 1.13 * fraction{ sin(37° ) }{ sin (90° ) } = 0.68 ; ;


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 = 1.13 ; ; b = 0.9 ; ; c = 0.68 ; ;

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

p = a+b+c = 1.13+0.9+0.68 = 2.7 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 2.7 }{ 2 } = 1.35 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1.35 * (1.35-1.13)(1.35-0.9)(1.35-0.68) } ; ; T = sqrt{ 0.09 } = 0.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.3 }{ 1.13 } = 0.54 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.3 }{ 0.9 } = 0.68 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.3 }{ 0.68 } = 0.9 ; ;

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{ 1.13**2-0.9**2-0.68**2 }{ 2 * 0.9 * 0.68 } ) = 90° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 0.9**2-1.13**2-0.68**2 }{ 2 * 1.13 * 0.68 } ) = 53° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 0.68**2-1.13**2-0.9**2 }{ 2 * 0.9 * 1.13 } ) = 37° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.3 }{ 1.35 } = 0.23 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1.13 }{ 2 * sin 90° } = 0.56 ; ;




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