17 21 24 triangle

Acute scalene triangle.

Sides: a = 17   b = 21   c = 24

Area: T = 174.2998594372
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 43.76217426927° = 43°45'42″ = 0.76437864964 rad
Angle ∠ B = β = 58.69440499332° = 58°41'39″ = 1.02444044227 rad
Angle ∠ C = γ = 77.54442073741° = 77°32'39″ = 1.35334017345 rad

Height: ha = 20.50657169849
Height: hb = 16.65998661307
Height: hc = 14.52548828643

Median: ma = 20.88765985742
Median: mb = 17.9511323071
Median: mc = 14.86660687473

Inradius: r = 5.62325353023
Circumradius: R = 12.28992557322

Vertex coordinates: A[24; 0] B[0; 0] C[8.83333333333; 14.52548828643]
Centroid: CG[10.94444444444; 4.84216276214]
Coordinates of the circumscribed circle: U[12; 2.65106237854]
Coordinates of the inscribed circle: I[10; 5.62325353023]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.2388257307° = 136°14'18″ = 0.76437864964 rad
∠ B' = β' = 121.3065950067° = 121°18'21″ = 1.02444044227 rad
∠ C' = γ' = 102.4565792626° = 102°27'21″ = 1.35334017345 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 = 17 ; ; b = 21 ; ; c = 24 ; ;

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

p = a+b+c = 17+21+24 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-17)(31-21)(31-24) } ; ; T = sqrt{ 30380 } = 174.3 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 174.3 }{ 17 } = 20.51 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 174.3 }{ 21 } = 16.6 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 174.3 }{ 24 } = 14.52 ; ;

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{ 17**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 43° 45'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-17**2-24**2 }{ 2 * 17 * 24 } ) = 58° 41'39" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-17**2-21**2 }{ 2 * 21 * 17 } ) = 77° 32'39" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 174.3 }{ 31 } = 5.62 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 17 }{ 2 * sin 43° 45'42" } = 12.29 ; ;




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