13 15 17 triangle

Acute scalene triangle.

Sides: a = 13   b = 15   c = 17

Area: T = 93.98998801916
Perimeter: p = 45
Semiperimeter: s = 22.5

Angle ∠ A = α = 47.43215457967° = 47°25'54″ = 0.82878366435 rad
Angle ∠ B = β = 58.18769524789° = 58°11'13″ = 1.01655539025 rad
Angle ∠ C = γ = 74.38215017244° = 74°22'53″ = 1.29882021077 rad

Height: ha = 14.44661354141
Height: hb = 12.52199840255
Height: hc = 11.04770447284

Median: ma = 14.65443508898
Median: mb = 13.14334394281
Median: mc = 11.16991539518

Inradius: r = 4.17333280085
Circumradius: R = 8.82658898553

Vertex coordinates: A[17; 0] B[0; 0] C[6.85329411765; 11.04770447284]
Centroid: CG[7.95109803922; 3.68223482428]
Coordinates of the circumscribed circle: U[8.5; 2.37662011149]
Coordinates of the inscribed circle: I[7.5; 4.17333280085]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 132.5688454203° = 132°34'6″ = 0.82878366435 rad
∠ B' = β' = 121.8133047521° = 121°48'47″ = 1.01655539025 rad
∠ C' = γ' = 105.6188498276° = 105°37'7″ = 1.29882021077 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 = 15 ; ; c = 17 ; ;

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

p = a+b+c = 13+15+17 = 45 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 45 }{ 2 } = 22.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 22.5 * (22.5-13)(22.5-15)(22.5-17) } ; ; T = sqrt{ 8817.19 } = 93.9 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 93.9 }{ 13 } = 14.45 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 93.9 }{ 15 } = 12.52 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 93.9 }{ 17 } = 11.05 ; ;

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**2-15**2-17**2 }{ 2 * 15 * 17 } ) = 47° 25'54" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 15**2-13**2-17**2 }{ 2 * 13 * 17 } ) = 58° 11'13" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 17**2-13**2-15**2 }{ 2 * 15 * 13 } ) = 74° 22'53" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 93.9 }{ 22.5 } = 4.17 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 47° 25'54" } = 8.83 ; ;




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