Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Right scalene triangle.

Sides: a = 3405   b = 912.3677000228   c = 3525.11553943

Area: T = 1553304.818789
Perimeter: p = 7842.482239452
Semiperimeter: s = 3921.241119726

Angle ∠ A = α = 75° = 1.3098996939 rad
Angle ∠ B = β = 15° = 0.26217993878 rad
Angle ∠ C = γ = 90° = 1.57107963268 rad

Height: ha = 912.3677000228
Height: hb = 3405
Height: hc = 881.2798848574

Median: ma = 1931.559890231
Median: mb = 3435.423259202
Median: mc = 1762.558769715

Inradius: r = 396.1265802966
Circumradius: R = 1762.558769715

Vertex coordinates: A[3525.11553943; 0] B[0; 0] C[3288.977743851; 881.2798848574]
Centroid: CG[2271.36442776; 293.7659616191]
Coordinates of the circumscribed circle: U[1762.558769715; 0]
Coordinates of the inscribed circle: I[3008.874419703; 396.1265802966]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 105° = 1.3098996939 rad
∠ B' = β' = 165° = 0.26217993878 rad
∠ C' = γ' = 90° = 1.57107963268 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 75° ; ; beta = 15° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 75° - 15° = 90° ; ;

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

a = 3405 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 3405 * fraction{ sin(15° ) }{ sin (75° ) } = 912.37 ; ;

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 = 3405 * fraction{ sin(90° ) }{ sin (75° ) } = 3525.12 ; ;


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 = 3405 ; ; b = 912.37 ; ; c = 3525.12 ; ;

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

p = a+b+c = 3405+912.37+3525.12 = 7842.48 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 7842.48 }{ 2 } = 3921.24 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 3921.24 * (3921.24-3405)(3921.24-912.37)(3921.24-3525.12) } ; ; T = sqrt{ 2.413 * 10**{ 12 } } = 1553304.82 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 1553304.82 }{ 3405 } = 912.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 1553304.82 }{ 912.37 } = 3405 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 1553304.82 }{ 3525.12 } = 881.28 ; ;

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{ 3405**2-912.37**2-3525.12**2 }{ 2 * 912.37 * 3525.12 } ) = 75° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 912.37**2-3405**2-3525.12**2 }{ 2 * 3405 * 3525.12 } ) = 15° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 3525.12**2-3405**2-912.37**2 }{ 2 * 912.37 * 3405 } ) = 90° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 1553304.82 }{ 3921.24 } = 396.13 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3405 }{ 2 * sin 75° } = 1762.56 ; ;




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