Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Acute scalene triangle.

Sides: a = 8.7   b = 7.3055314688   c = 8.91101527297

Area: T = 29.25219126745
Perimeter: p = 24.91554674177
Semiperimeter: s = 12.45877337088

Angle ∠ A = α = 64° = 1.11770107213 rad
Angle ∠ B = β = 49° = 0.85552113335 rad
Angle ∠ C = γ = 67° = 1.16993705988 rad

Height: ha = 6.72545776263
Height: hb = 8.0088392225
Height: hc = 6.56659733479

Median: ma = 6.88988839574
Median: mb = 8.01223969672
Median: mc = 6.68443927121

Inradius: r = 2.34880926273
Circumradius: R = 4.84398184411

Vertex coordinates: A[8.91101527297; 0] B[0; 0] C[5.70877135522; 6.56659733479]
Centroid: CG[4.8732622094; 2.18986577826]
Coordinates of the circumscribed circle: U[4.45550763648; 1.89110677212]
Coordinates of the inscribed circle: I[5.15224190208; 2.34880926273]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 116° = 1.11770107213 rad
∠ B' = β' = 131° = 0.85552113335 rad
∠ C' = γ' = 113° = 1.16993705988 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 64° ; ; beta = 49° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 64° - 49° = 67° ; ;

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

a = 8.7 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 8.7 * fraction{ sin(49° ) }{ sin (64° ) } = 7.31 ; ;

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 = 8.7 * fraction{ sin(67° ) }{ sin (64° ) } = 8.91 ; ;


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 = 8.7 ; ; b = 7.31 ; ; c = 8.91 ; ;

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

p = a+b+c = 8.7+7.31+8.91 = 24.92 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 24.92 }{ 2 } = 12.46 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 12.46 * (12.46-8.7)(12.46-7.31)(12.46-8.91) } ; ; T = sqrt{ 855.67 } = 29.25 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 29.25 }{ 8.7 } = 6.72 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 29.25 }{ 7.31 } = 8.01 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 29.25 }{ 8.91 } = 6.57 ; ;

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{ 8.7**2-7.31**2-8.91**2 }{ 2 * 7.31 * 8.91 } ) = 64° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 7.31**2-8.7**2-8.91**2 }{ 2 * 8.7 * 8.91 } ) = 49° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.91**2-8.7**2-7.31**2 }{ 2 * 7.31 * 8.7 } ) = 67° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 29.25 }{ 12.46 } = 2.35 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8.7 }{ 2 * sin 64° } = 4.84 ; ;




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