Triangle calculator AAS

Please enter two angles and one opposite side
°
°


Obtuse scalene triangle.

Sides: a = 46   b = 35.58107608044   c = 19.78879619336

Area: T = 332.85659817759
Perimeter: p = 101.3698722738
Semiperimeter: s = 50.6844361369

Angle ∠ A = α = 109° = 1.90224088847 rad
Angle ∠ B = β = 47° = 0.82203047484 rad
Angle ∠ C = γ = 24° = 0.41988790205 rad

Height: ha = 14.47219992076
Height: hb = 18.71098855815
Height: hc = 33.64222702745

Median: ma = 17.314406909
Median: mb = 30.61550956864
Median: mc = 39.91437120591

Inradius: r = 6.5677232432
Circumradius: R = 24.32552756673

Vertex coordinates: A[19.78879619336; 0] B[0; 0] C[31.37219245629; 33.64222702745]
Centroid: CG[17.05332954988; 11.21440900915]
Coordinates of the circumscribed circle: U[9.89439809668; 22.22222450918]
Coordinates of the inscribed circle: I[15.10436005646; 6.5677232432]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 71° = 1.90224088847 rad
∠ B' = β' = 133° = 0.82203047484 rad
∠ C' = γ' = 156° = 0.41988790205 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 109° ; ; beta = 47° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 109° - 47° = 24° ; ;

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

a = 46 ; ; ; ; fraction{ b }{ a } = fraction{ sin beta }{ sin alpha } ; ; ; ; b = a * fraction{ sin beta }{ sin alpha } ; ; ; ; b = 46 * fraction{ sin 47° }{ sin 109° } = 35.58 ; ;

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 = 46 * fraction{ sin 24° }{ sin 109° } = 19.79 ; ;


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 = 46 ; ; b = 35.58 ; ; c = 19.79 ; ;

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

p = a+b+c = 46+35.58+19.79 = 101.37 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 101.37 }{ 2 } = 50.68 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 50.68 * (50.68-46)(50.68-35.58)(50.68-19.79) } ; ; T = sqrt{ 110793.1 } = 332.86 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 332.86 }{ 46 } = 14.47 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 332.86 }{ 35.58 } = 18.71 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 332.86 }{ 19.79 } = 33.64 ; ;

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{ b**2+c**2-a**2 }{ 2bc } ) = arccos( fraction{ 35.58**2+19.79**2-46**2 }{ 2 * 35.58 * 19.79 } ) = 109° ; ; beta = arccos( fraction{ a**2+c**2-b**2 }{ 2ac } ) = arccos( fraction{ 46**2+19.79**2-35.58**2 }{ 2 * 46 * 19.79 } ) = 47° ; ; gamma = 180° - alpha - beta = 180° - 109° - 47° = 24° ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 332.86 }{ 50.68 } = 6.57 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 46 }{ 2 * sin 109° } = 24.33 ; ;

11. Calculation of medians

m_a = fraction{ sqrt{ 2 b**2+2c**2 - a**2 } }{ 2 } = fraction{ sqrt{ 2 * 35.58**2+2 * 19.79**2 - 46**2 } }{ 2 } = 17.314 ; ; m_b = fraction{ sqrt{ 2 c**2+2a**2 - b**2 } }{ 2 } = fraction{ sqrt{ 2 * 19.79**2+2 * 46**2 - 35.58**2 } }{ 2 } = 30.615 ; ; m_c = fraction{ sqrt{ 2 b**2+2a**2 - c**2 } }{ 2 } = fraction{ sqrt{ 2 * 35.58**2+2 * 46**2 - 19.79**2 } }{ 2 } = 39.914 ; ;
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