Triangle calculator AAS

Please enter two angles and one opposite side
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°


Acute scalene triangle.

Sides: a = 44   b = 44.92993711216   c = 60.25551339805

Area: T = 985.1222402573
Perimeter: p = 149.1854505102
Semiperimeter: s = 74.59222525511

Angle ∠ A = α = 46.7° = 46°42' = 0.81550687607 rad
Angle ∠ B = β = 48° = 0.8387758041 rad
Angle ∠ C = γ = 85.3° = 85°18' = 1.4898765852 rad

Height: ha = 44.77882910261
Height: hb = 43.852204502
Height: hc = 32.6988372321

Median: ma = 48.38804173214
Median: mb = 47.73655055295
Median: mc = 32.70655637765

Inradius: r = 13.20767657013
Circumradius: R = 30.2299216151

Vertex coordinates: A[60.25551339805; 0] B[0; 0] C[29.44217466798; 32.6988372321]
Centroid: CG[29.89989602201; 10.89994574403]
Coordinates of the circumscribed circle: U[30.12875669903; 2.47769368885]
Coordinates of the inscribed circle: I[29.66328814294; 13.20767657013]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 133.3° = 133°18' = 0.81550687607 rad
∠ B' = β' = 132° = 0.8387758041 rad
∠ C' = γ' = 94.7° = 94°42' = 1.4898765852 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 46° 42' ; ; beta = 48° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 46° 42' - 48° = 85° 18' ; ;

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

a = 44 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 44 * fraction{ sin(48° ) }{ sin (46° 42') } = 44.93 ; ;

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 = 44 * fraction{ sin(85° 18') }{ sin (46° 42') } = 60.26 ; ;


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 = 44 ; ; b = 44.93 ; ; c = 60.26 ; ;

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

p = a+b+c = 44+44.93+60.26 = 149.18 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 149.18 }{ 2 } = 74.59 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 74.59 * (74.59-44)(74.59-44.93)(74.59-60.26) } ; ; T = sqrt{ 970466.15 } = 985.12 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 985.12 }{ 44 } = 44.78 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 985.12 }{ 44.93 } = 43.85 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 985.12 }{ 60.26 } = 32.7 ; ;

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{ 44**2-44.93**2-60.26**2 }{ 2 * 44.93 * 60.26 } ) = 46° 42' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 44.93**2-44**2-60.26**2 }{ 2 * 44 * 60.26 } ) = 48° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 60.26**2-44**2-44.93**2 }{ 2 * 44.93 * 44 } ) = 85° 18' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 985.12 }{ 74.59 } = 13.21 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 44 }{ 2 * sin 46° 42' } = 30.23 ; ;




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