Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Acute scalene triangle.

Sides: a = 7.8   b = 10.13882949686   c = 8.80326320116

Area: T = 33.16604894372
Perimeter: p = 26.74109269801
Semiperimeter: s = 13.37704634901

Angle ∠ A = α = 48° = 0.8387758041 rad
Angle ∠ B = β = 75° = 1.3098996939 rad
Angle ∠ C = γ = 57° = 0.99548376736 rad

Height: ha = 8.50326895993
Height: hb = 6.542163043
Height: hc = 7.53442214451

Median: ma = 8.65659619685
Median: mb = 6.59329438757
Median: mc = 7.90219573431

Inradius: r = 2.48801301362
Circumradius: R = 5.24879676455

Vertex coordinates: A[8.80326320116; 0] B[0; 0] C[2.01987885518; 7.53442214451]
Centroid: CG[3.60771401878; 2.51114071484]
Coordinates of the circumscribed circle: U[4.40113160058; 2.85882480342]
Coordinates of the inscribed circle: I[3.23221685215; 2.48801301362]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132° = 0.8387758041 rad
∠ B' = β' = 105° = 1.3098996939 rad
∠ C' = γ' = 123° = 0.99548376736 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 48° ; ; beta = 75° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 48° - 75° = 57° ; ;

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

a = 7.8 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 7.8 * fraction{ sin(75° ) }{ sin (48° ) } = 10.14 ; ;

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 = 7.8 * fraction{ sin(57° ) }{ sin (48° ) } = 8.8 ; ;


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 = 7.8 ; ; b = 10.14 ; ; c = 8.8 ; ;

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

p = a+b+c = 7.8+10.14+8.8 = 26.74 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 26.74 }{ 2 } = 13.37 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 13.37 * (13.37-7.8)(13.37-10.14)(13.37-8.8) } ; ; T = sqrt{ 1099.62 } = 33.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 33.16 }{ 7.8 } = 8.5 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 33.16 }{ 10.14 } = 6.54 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 33.16 }{ 8.8 } = 7.53 ; ;

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{ 7.8**2-10.14**2-8.8**2 }{ 2 * 10.14 * 8.8 } ) = 48° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 10.14**2-7.8**2-8.8**2 }{ 2 * 7.8 * 8.8 } ) = 75° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 8.8**2-7.8**2-10.14**2 }{ 2 * 10.14 * 7.8 } ) = 57° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 33.16 }{ 13.37 } = 2.48 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 7.8 }{ 2 * sin 48° } = 5.25 ; ;




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