13 17 24 triangle

Obtuse scalene triangle.

Sides: a = 13   b = 17   c = 24

Area: T = 106.4899436096
Perimeter: p = 54
Semiperimeter: s = 27

Angle ∠ A = α = 31.46769762933° = 31°28'1″ = 0.5499202342 rad
Angle ∠ B = β = 43.04990798002° = 43°2'57″ = 0.75113481825 rad
Angle ∠ C = γ = 105.4843943906° = 105°29'2″ = 1.84110421292 rad

Height: ha = 16.38329901686
Height: hb = 12.52881689524
Height: hc = 8.87441196746

Median: ma = 19.75547462651
Median: mb = 17.32877234512
Median: mc = 9.22195444573

Inradius: r = 3.94440531887
Circumradius: R = 12.4521939353

Vertex coordinates: A[24; 0] B[0; 0] C[9.5; 8.87441196746]
Centroid: CG[11.16766666667; 2.95880398915]
Coordinates of the circumscribed circle: U[12; -3.32442734019]
Coordinates of the inscribed circle: I[10; 3.94440531887]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.5333023707° = 148°31'59″ = 0.5499202342 rad
∠ B' = β' = 136.95109202° = 136°57'3″ = 0.75113481825 rad
∠ C' = γ' = 74.51660560935° = 74°30'58″ = 1.84110421292 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 = 17 ; ; c = 24 ; ;

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

p = a+b+c = 13+17+24 = 54 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 54 }{ 2 } = 27 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 27 * (27-13)(27-17)(27-24) } ; ; T = sqrt{ 11340 } = 106.49 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 106.49 }{ 13 } = 16.38 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 106.49 }{ 17 } = 12.53 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 106.49 }{ 24 } = 8.87 ; ;

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**2-17**2-24**2 }{ 2 * 17 * 24 } ) = 31° 28'1" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 17**2-13**2-24**2 }{ 2 * 13 * 24 } ) = 43° 2'57" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-13**2-17**2 }{ 2 * 17 * 13 } ) = 105° 29'2" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 106.49 }{ 27 } = 3.94 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 31° 28'1" } = 12.45 ; ;




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