24 28 29 triangle

Acute scalene triangle.

Sides: a = 24   b = 28   c = 29

Area: T = 309.9376989564
Perimeter: p = 81
Semiperimeter: s = 40.5

Angle ∠ A = α = 49.76441130987° = 49°45'51″ = 0.86985476229 rad
Angle ∠ B = β = 62.95217046453° = 62°57'6″ = 1.09987145158 rad
Angle ∠ C = γ = 67.2844182256° = 67°17'3″ = 1.17443305149 rad

Height: ha = 25.82880824637
Height: hb = 22.13883563974
Height: hc = 21.37549647975

Median: ma = 25.85553669477
Median: mb = 22.63884628453
Median: mc = 21.67437168017

Inradius: r = 7.65327651744
Circumradius: R = 15.71993241338

Vertex coordinates: A[29; 0] B[0; 0] C[10.91437931034; 21.37549647975]
Centroid: CG[13.30545977011; 7.12549882658]
Coordinates of the circumscribed circle: U[14.5; 6.07701854356]
Coordinates of the inscribed circle: I[12.5; 7.65327651744]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 130.2365886901° = 130°14'9″ = 0.86985476229 rad
∠ B' = β' = 117.0488295355° = 117°2'54″ = 1.09987145158 rad
∠ C' = γ' = 112.7165817744° = 112°42'57″ = 1.17443305149 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 = 24 ; ; b = 28 ; ; c = 29 ; ;

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

p = a+b+c = 24+28+29 = 81 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 81 }{ 2 } = 40.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 40.5 * (40.5-24)(40.5-28)(40.5-29) } ; ; T = sqrt{ 96060.94 } = 309.94 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 309.94 }{ 24 } = 25.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 309.94 }{ 28 } = 22.14 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 309.94 }{ 29 } = 21.37 ; ;

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{ 24**2-28**2-29**2 }{ 2 * 28 * 29 } ) = 49° 45'51" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 28**2-24**2-29**2 }{ 2 * 24 * 29 } ) = 62° 57'6" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-24**2-28**2 }{ 2 * 28 * 24 } ) = 67° 17'3" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 309.94 }{ 40.5 } = 7.65 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 24 }{ 2 * sin 49° 45'51" } = 15.72 ; ;




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