19 27 29 triangle

Acute scalene triangle.

Sides: a = 19   b = 27   c = 29

Area: T = 248.8321644893
Perimeter: p = 75
Semiperimeter: s = 37.5

Angle ∠ A = α = 39.46334085924° = 39°27'48″ = 0.6898766414 rad
Angle ∠ B = β = 64.58219557631° = 64°34'55″ = 1.12771677654 rad
Angle ∠ C = γ = 75.95546356446° = 75°57'17″ = 1.32656584742 rad

Height: ha = 26.19328047255
Height: hb = 18.43219736958
Height: hc = 17.1610803096

Median: ma = 26.35881107062
Median: mb = 20.46333819297
Median: mc = 18.29661744635

Inradius: r = 6.63655105305
Circumradius: R = 14.94768529278

Vertex coordinates: A[29; 0] B[0; 0] C[8.15551724138; 17.1610803096]
Centroid: CG[12.38550574713; 5.72202676987]
Coordinates of the circumscribed circle: U[14.5; 3.62774526111]
Coordinates of the inscribed circle: I[10.5; 6.63655105305]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 140.5376591408° = 140°32'12″ = 0.6898766414 rad
∠ B' = β' = 115.4188044237° = 115°25'5″ = 1.12771677654 rad
∠ C' = γ' = 104.0455364355° = 104°2'43″ = 1.32656584742 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 = 27 ; ; c = 29 ; ;

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

p = a+b+c = 19+27+29 = 75 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 75 }{ 2 } = 37.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37.5 * (37.5-19)(37.5-27)(37.5-29) } ; ; T = sqrt{ 61917.19 } = 248.83 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 248.83 }{ 19 } = 26.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 248.83 }{ 27 } = 18.43 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 248.83 }{ 29 } = 17.16 ; ;

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**2-27**2-29**2 }{ 2 * 27 * 29 } ) = 39° 27'48" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-19**2-29**2 }{ 2 * 19 * 29 } ) = 64° 34'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 29**2-19**2-27**2 }{ 2 * 27 * 19 } ) = 75° 57'17" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 248.83 }{ 37.5 } = 6.64 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 39° 27'48" } = 14.95 ; ;




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