13 23 29 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 23   c = 29

Area: T = 145.1632624322
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 25.80224907766° = 25°48'9″ = 0.45503384193 rad
Angle ∠ B = β = 50.36222856093° = 50°21'44″ = 0.87989877027 rad
Angle ∠ C = γ = 103.8355223614° = 103°50'7″ = 1.81222665316 rad

Height: ha = 22.33327114341
Height: hb = 12.62328368976
Height: hc = 10.01112154705

Median: ma = 25.35325146682
Median: mb = 19.30767345763
Median: mc = 11.77992189894

Inradius: r = 4.46765422868
Circumradius: R = 14.93332516557

Vertex coordinates: A[29; 0] B[0; 0] C[8.29331034483; 10.01112154705]
Centroid: CG[12.43110344828; 3.33770718235]
Coordinates of the circumscribed circle: U[14.5; -3.57109949611]
Coordinates of the inscribed circle: I[9.5; 4.46765422868]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 154.1987509223° = 154°11'51″ = 0.45503384193 rad
∠ B' = β' = 129.6387714391° = 129°38'16″ = 0.87989877027 rad
∠ C' = γ' = 76.1654776386° = 76°9'53″ = 1.81222665316 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 = 29 ; ;

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

p = a+b+c = 13+23+29 = 65 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 65 }{ 2 } = 32.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.5 * (32.5-13)(32.5-23)(32.5-29) } ; ; T = sqrt{ 21072.19 } = 145.16 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 145.16 }{ 13 } = 22.33 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 145.16 }{ 23 } = 12.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 145.16 }{ 29 } = 10.01 ; ;

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**2-29**2 }{ 2 * 23 * 29 } ) = 25° 48'9" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 23**2-13**2-29**2 }{ 2 * 13 * 29 } ) = 50° 21'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-13**2-23**2 }{ 2 * 23 * 13 } ) = 103° 50'7" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 145.16 }{ 32.5 } = 4.47 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 25° 48'9" } = 14.93 ; ;




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