Triangle calculator AAS

Please enter two angles and one opposite side
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°


Acute scalene triangle.

Sides: a = 1049   b = 1409.045472999   c = 1506.976563294

Area: T = 710414.651121
Perimeter: p = 3965.022036294
Semiperimeter: s = 1982.511018147

Angle ∠ A = α = 42° = 0.73330382858 rad
Angle ∠ B = β = 64° = 1.11770107213 rad
Angle ∠ C = γ = 74° = 1.29215436465 rad

Height: ha = 1354.461072681
Height: hb = 1008.364351904
Height: hc = 942.8354954568

Median: ma = 1361.283287094
Median: mb = 1090.567706184
Median: mc = 987.5021967588

Inradius: r = 358.3410985005
Circumradius: R = 783.8532950404

Vertex coordinates: A[1506.976563294; 0] B[0; 0] C[459.8511332982; 942.8354954568]
Centroid: CG[655.6098988642; 314.2788318189]
Coordinates of the circumscribed circle: U[753.4887816472; 216.0599154599]
Coordinates of the inscribed circle: I[573.4655451475; 358.3410985005]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 138° = 0.73330382858 rad
∠ B' = β' = 116° = 1.11770107213 rad
∠ C' = γ' = 106° = 1.29215436465 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 42° ; ; beta = 64° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 42° - 64° = 74° ; ;

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

a = 1049 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 1049 * fraction{ sin(64° ) }{ sin (42° ) } = 1409.04 ; ;

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 = 1049 * fraction{ sin(74° ) }{ sin (42° ) } = 1506.98 ; ;


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 = 1049 ; ; b = 1409.04 ; ; c = 1506.98 ; ;

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

p = a+b+c = 1049+1409.04+1506.98 = 3965.02 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 3965.02 }{ 2 } = 1982.51 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 1982.51 * (1982.51-1049)(1982.51-1409.04)(1982.51-1506.98) } ; ; T = sqrt{ 504688976654 } = 710414.65 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 710414.65 }{ 1049 } = 1354.46 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 710414.65 }{ 1409.04 } = 1008.36 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 710414.65 }{ 1506.98 } = 942.83 ; ;

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{ 1049**2-1409.04**2-1506.98**2 }{ 2 * 1409.04 * 1506.98 } ) = 42° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 1409.04**2-1049**2-1506.98**2 }{ 2 * 1049 * 1506.98 } ) = 64° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 1506.98**2-1049**2-1409.04**2 }{ 2 * 1409.04 * 1049 } ) = 74° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 710414.65 }{ 1982.51 } = 358.34 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 1049 }{ 2 * sin 42° } = 783.85 ; ;




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