13 21 24 triangle

Acute scalene triangle.

Sides: a = 13   b = 21   c = 24

Area: T = 136.2355090927
Perimeter: p = 58
Semiperimeter: s = 29

Angle ∠ A = α = 32.72655443586° = 32°43'32″ = 0.57111684986 rad
Angle ∠ B = β = 60.84546345737° = 60°50'41″ = 1.06219392055 rad
Angle ∠ C = γ = 86.43298210677° = 86°25'47″ = 1.50884849495 rad

Height: ha = 20.95992447581
Height: hb = 12.97547705645
Height: hc = 11.3532924244

Median: ma = 21.59328228817
Median: mb = 16.19441347407
Median: mc = 12.68985775404

Inradius: r = 4.69877617561
Circumradius: R = 12.02333339946

Vertex coordinates: A[24; 0] B[0; 0] C[6.33333333333; 11.3532924244]
Centroid: CG[10.11111111111; 3.78443080813]
Coordinates of the circumscribed circle: U[12; 0.74987057799]
Coordinates of the inscribed circle: I[8; 4.69877617561]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 147.2744455641° = 147°16'28″ = 0.57111684986 rad
∠ B' = β' = 119.1555365426° = 119°9'19″ = 1.06219392055 rad
∠ C' = γ' = 93.57701789323° = 93°34'13″ = 1.50884849495 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 = 21 ; ; c = 24 ; ;

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

p = a+b+c = 13+21+24 = 58 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 58 }{ 2 } = 29 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 29 * (29-13)(29-21)(29-24) } ; ; T = sqrt{ 18560 } = 136.24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 136.24 }{ 13 } = 20.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 136.24 }{ 21 } = 12.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 136.24 }{ 24 } = 11.35 ; ;

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**2-21**2-24**2 }{ 2 * 21 * 24 } ) = 32° 43'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 21**2-13**2-24**2 }{ 2 * 13 * 24 } ) = 60° 50'41" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-13**2-21**2 }{ 2 * 21 * 13 } ) = 86° 25'47" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 136.24 }{ 29 } = 4.7 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 32° 43'32" } = 12.02 ; ;




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