Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 114.88   b = 121.4787579374   c = 39.4899085713

Area: T = 2268.253308336
Perimeter: p = 275.8476665087
Semiperimeter: s = 137.9233332543

Angle ∠ A = α = 71.03° = 71°1'48″ = 1.24397073677 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 18.97° = 18°58'12″ = 0.33110889591 rad

Height: ha = 39.4899085713
Height: hb = 37.34443905461
Height: hc = 114.88

Median: ma = 69.7054673376
Median: mb = 60.73987896868
Median: mc = 116.5644408687

Inradius: r = 16.44657531698
Circumradius: R = 60.73987896868

Vertex coordinates: A[39.4899085713; 0] B[0; 0] C[0; 114.88]
Centroid: CG[13.1633028571; 38.29333333333]
Coordinates of the circumscribed circle: U[19.74545428565; 57.44]
Coordinates of the inscribed circle: I[16.44657531698; 16.44657531698]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 108.97° = 108°58'12″ = 1.24397073677 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 161.03° = 161°1'48″ = 0.33110889591 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 71° 1'48" ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 71° 1'48" - 90° = 18° 58'12" ; ;

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

a = 114.88 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 114.88 * fraction{ sin(90° ) }{ sin (71° 1'48") } = 121.48 ; ;

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 = 114.88 * fraction{ sin(18° 58'12") }{ sin (71° 1'48") } = 39.49 ; ;


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 = 114.88 ; ; b = 121.48 ; ; c = 39.49 ; ;

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

p = a+b+c = 114.88+121.48+39.49 = 275.85 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 275.85 }{ 2 } = 137.92 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 137.92 * (137.92-114.88)(137.92-121.48)(137.92-39.49) } ; ; T = sqrt{ 5144972.05 } = 2268.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 2268.25 }{ 114.88 } = 39.49 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 2268.25 }{ 121.48 } = 37.34 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 2268.25 }{ 39.49 } = 114.88 ; ;

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{ 114.88**2-121.48**2-39.49**2 }{ 2 * 121.48 * 39.49 } ) = 71° 1'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 121.48**2-114.88**2-39.49**2 }{ 2 * 114.88 * 39.49 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 39.49**2-114.88**2-121.48**2 }{ 2 * 121.48 * 114.88 } ) = 18° 58'12" ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 2268.25 }{ 137.92 } = 16.45 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 114.88 }{ 2 * sin 71° 1'48" } = 60.74 ; ;

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