19 22 24 triangle

Acute scalene triangle.

Sides: a = 19   b = 22   c = 24

Area: T = 197.8854909733
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 48.55326288331° = 48°33'9″ = 0.84774032336 rad
Angle ∠ B = β = 60.21773922406° = 60°13'3″ = 1.05109917616 rad
Angle ∠ C = γ = 71.23299789263° = 71°13'48″ = 1.24331976584 rad

Height: ha = 20.83299904982
Height: hb = 17.99895372484
Height: hc = 16.49904091444

Median: ma = 20.97702169755
Median: mb = 18.64113518823
Median: mc = 16.68883192683

Inradius: r = 6.08987664533
Circumradius: R = 12.67440336258

Vertex coordinates: A[24; 0] B[0; 0] C[9.43875; 16.49904091444]
Centroid: CG[11.14658333333; 5.49768030481]
Coordinates of the circumscribed circle: U[12; 4.07881280447]
Coordinates of the inscribed circle: I[10.5; 6.08987664533]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 131.4477371167° = 131°26'51″ = 0.84774032336 rad
∠ B' = β' = 119.7832607759° = 119°46'57″ = 1.05109917616 rad
∠ C' = γ' = 108.7770021074° = 108°46'12″ = 1.24331976584 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 = 22 ; ; c = 24 ; ;

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

p = a+b+c = 19+22+24 = 65 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 65 }{ 2 } = 32.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 32.5 * (32.5-19)(32.5-22)(32.5-24) } ; ; T = sqrt{ 39158.44 } = 197.88 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 197.88 }{ 19 } = 20.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 197.88 }{ 22 } = 17.99 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 197.88 }{ 24 } = 16.49 ; ;

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**2-22**2-24**2 }{ 2 * 22 * 24 } ) = 48° 33'9" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 22**2-19**2-24**2 }{ 2 * 19 * 24 } ) = 60° 13'3" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 24**2-19**2-22**2 }{ 2 * 22 * 19 } ) = 71° 13'48" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 197.88 }{ 32.5 } = 6.09 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 48° 33'9" } = 12.67 ; ;




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