19 24 29 triangle

Acute scalene triangle.

Sides: a = 19   b = 24   c = 29

Area: T = 226.7333323532
Perimeter: p = 72
Semiperimeter: s = 36

Angle ∠ A = α = 40.6577248638° = 40°39'26″ = 0.71096028535 rad
Angle ∠ B = β = 55.38550797811° = 55°23'6″ = 0.96766519987 rad
Angle ∠ C = γ = 83.95876715809° = 83°57'28″ = 1.46553378014 rad

Height: ha = 23.8676665635
Height: hb = 18.89444436277
Height: hc = 15.63767809333

Median: ma = 24.86546335183
Median: mb = 21.37875583264
Median: mc = 16.077015868

Inradius: r = 6.29881478759
Circumradius: R = 14.58110062169

Vertex coordinates: A[29; 0] B[0; 0] C[10.79331034483; 15.63767809333]
Centroid: CG[13.26443678161; 5.21222603111]
Coordinates of the circumscribed circle: U[14.5; 1.53548427597]
Coordinates of the inscribed circle: I[12; 6.29881478759]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 139.3432751362° = 139°20'34″ = 0.71096028535 rad
∠ B' = β' = 124.6154920219° = 124°36'54″ = 0.96766519987 rad
∠ C' = γ' = 96.04223284191° = 96°2'32″ = 1.46553378014 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 = 19 ; ; b = 24 ; ; c = 29 ; ;

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

p = a+b+c = 19+24+29 = 72 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 72 }{ 2 } = 36 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36 * (36-19)(36-24)(36-29) } ; ; T = sqrt{ 51408 } = 226.73 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 226.73 }{ 19 } = 23.87 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 226.73 }{ 24 } = 18.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 226.73 }{ 29 } = 15.64 ; ;

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{ 19**2-24**2-29**2 }{ 2 * 24 * 29 } ) = 40° 39'26" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-19**2-29**2 }{ 2 * 19 * 29 } ) = 55° 23'6" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-19**2-24**2 }{ 2 * 24 * 19 } ) = 83° 57'28" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 226.73 }{ 36 } = 6.3 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 40° 39'26" } = 14.58 ; ;




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