13 23 30 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 23   c = 30

Area: T = 140.7122472795
Perimeter: p = 66
Semiperimeter: s = 33

Angle ∠ A = α = 24.07106146226° = 24°4'14″ = 0.42201114781 rad
Angle ∠ B = β = 46.18769385396° = 46°11'13″ = 0.80661141489 rad
Angle ∠ C = γ = 109.7422446838° = 109°44'33″ = 1.91553670265 rad

Height: ha = 21.64880727376
Height: hb = 12.23658671995
Height: hc = 9.38108315196

Median: ma = 25.92877843249
Median: mb = 20.05661711201
Median: mc = 11.13655287257

Inradius: r = 4.26440143271
Circumradius: R = 15.93767535476

Vertex coordinates: A[30; 0] B[0; 0] C[9; 9.38108315196]
Centroid: CG[13; 3.12769438399]
Coordinates of the circumscribed circle: U[15; -5.3833318088]
Coordinates of the inscribed circle: I[10; 4.26440143271]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 155.9299385377° = 155°55'46″ = 0.42201114781 rad
∠ B' = β' = 133.813306146° = 133°48'47″ = 0.80661141489 rad
∠ C' = γ' = 70.25875531622° = 70°15'27″ = 1.91553670265 rad

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How did we calculate this triangle?

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 = 13 ; ; b = 23 ; ; c = 30 ; ;

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

p = a+b+c = 13+23+30 = 66 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 66 }{ 2 } = 33 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33 * (33-13)(33-23)(33-30) } ; ; T = sqrt{ 19800 } = 140.71 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 140.71 }{ 13 } = 21.65 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 140.71 }{ 23 } = 12.24 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 140.71 }{ 30 } = 9.38 ; ;

5. 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{ 13**2-23**2-30**2 }{ 2 * 23 * 30 } ) = 24° 4'14" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-13**2-30**2 }{ 2 * 13 * 30 } ) = 46° 11'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 30**2-13**2-23**2 }{ 2 * 23 * 13 } ) = 109° 44'33" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 140.71 }{ 33 } = 4.26 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 24° 4'14" } = 15.94 ; ;




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