Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse isosceles triangle.

Sides: a = 300   b = 183.1166188314   c = 183.1166188314

Area: T = 15754.67696097
Perimeter: p = 666.2322376628
Semiperimeter: s = 333.1166188314

Angle ∠ A = α = 110° = 1.92198621772 rad
Angle ∠ B = β = 35° = 0.61108652382 rad
Angle ∠ C = γ = 35° = 0.61108652382 rad

Height: ha = 105.0311130731
Height: hb = 172.0732930905
Height: hc = 172.0732930905

Median: ma = 105.0311130731
Median: mb = 231.0477364421
Median: mc = 231.0477364421

Inradius: r = 47.29548183318
Circumradius: R = 159.6276665871

Vertex coordinates: A[183.1166188314; 0] B[0; 0] C[245.7465613287; 172.0732930905]
Centroid: CG[142.9543933867; 57.35876436351]
Coordinates of the circumscribed circle: U[91.55880941571; 130.7598509672]
Coordinates of the inscribed circle: I[150; 47.29548183318]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 70° = 1.92198621772 rad
∠ B' = β' = 145° = 0.61108652382 rad
∠ C' = γ' = 145° = 0.61108652382 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 110° ; ; beta = 35° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 110° - 35° = 35° ; ;

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

a = 300 ; ; ; ; fraction{ b }{ a } = fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = a * fraction{ sin( beta ) }{ sin ( alpha ) } ; ; ; ; b = 300 * fraction{ sin(35° ) }{ sin (110° ) } = 183.12 ; ;

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 = 300 * fraction{ sin(35° ) }{ sin (110° ) } = 183.12 ; ;


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 = 300 ; ; b = 183.12 ; ; c = 183.12 ; ;

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

p = a+b+c = 300+183.12+183.12 = 666.23 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 666.23 }{ 2 } = 333.12 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 333.12 * (333.12-300)(333.12-183.12)(333.12-183.12) } ; ; T = sqrt{ 248209614.51 } = 15754.67 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 15754.67 }{ 300 } = 105.03 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 15754.67 }{ 183.12 } = 172.07 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 15754.67 }{ 183.12 } = 172.07 ; ;

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{ 300**2-183.12**2-183.12**2 }{ 2 * 183.12 * 183.12 } ) = 110° ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 183.12**2-300**2-183.12**2 }{ 2 * 300 * 183.12 } ) = 35° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 183.12**2-300**2-183.12**2 }{ 2 * 183.12 * 300 } ) = 35° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 15754.67 }{ 333.12 } = 47.29 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 300 }{ 2 * sin 110° } = 159.63 ; ;




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