13 20 23 triangle

Acute scalene triangle.

Sides: a = 13   b = 20   c = 23

Area: T = 129.6154813968
Perimeter: p = 56
Semiperimeter: s = 28

Angle ∠ A = α = 34.30111527568° = 34°18'4″ = 0.59986680528 rad
Angle ∠ B = β = 60.1110573029° = 60°6'38″ = 1.04991274146 rad
Angle ∠ C = γ = 85.58882742142° = 85°35'18″ = 1.49437971861 rad

Height: ha = 19.94107406105
Height: hb = 12.96114813968
Height: hc = 11.27108533885

Median: ma = 20.5498722588
Median: mb = 15.78797338381
Median: mc = 12.33989626793

Inradius: r = 4.62991004989
Circumradius: R = 11.53441754097

Vertex coordinates: A[23; 0] B[0; 0] C[6.47882608696; 11.27108533885]
Centroid: CG[9.82660869565; 3.75769511295]
Coordinates of the circumscribed circle: U[11.5; 0.88772442623]
Coordinates of the inscribed circle: I[8; 4.62991004989]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 145.6998847243° = 145°41'56″ = 0.59986680528 rad
∠ B' = β' = 119.8899426971° = 119°53'22″ = 1.04991274146 rad
∠ C' = γ' = 94.41217257858° = 94°24'42″ = 1.49437971861 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 = 20 ; ; c = 23 ; ;

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

p = a+b+c = 13+20+23 = 56 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 56 }{ 2 } = 28 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28 * (28-13)(28-20)(28-23) } ; ; T = sqrt{ 16800 } = 129.61 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 129.61 }{ 13 } = 19.94 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 129.61 }{ 20 } = 12.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 129.61 }{ 23 } = 11.27 ; ;

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**2-20**2-23**2 }{ 2 * 20 * 23 } ) = 34° 18'4" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 20**2-13**2-23**2 }{ 2 * 13 * 23 } ) = 60° 6'38" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 23**2-13**2-20**2 }{ 2 * 20 * 13 } ) = 85° 35'18" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 129.61 }{ 28 } = 4.63 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 34° 18'4" } = 11.53 ; ;




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