19 24 27 triangle

Acute scalene triangle.

Sides: a = 19   b = 24   c = 27

Area: T = 221.9910990808
Perimeter: p = 70
Semiperimeter: s = 35

Angle ∠ A = α = 43.24879853748° = 43°14'53″ = 0.75548197396 rad
Angle ∠ B = β = 59.93554961114° = 59°56'8″ = 1.0466071746 rad
Angle ∠ C = γ = 76.81765185138° = 76°48'59″ = 1.3410701168 rad

Height: ha = 23.36774727167
Height: hb = 18.4999249234
Height: hc = 16.44437770969

Median: ma = 23.71218114028
Median: mb = 20.02549843945
Median: mc = 16.91989243157

Inradius: r = 6.34325997374
Circumradius: R = 13.86554275509

Vertex coordinates: A[27; 0] B[0; 0] C[9.51985185185; 16.44437770969]
Centroid: CG[12.17328395062; 5.48112590323]
Coordinates of the circumscribed circle: U[13.5; 3.16222904941]
Coordinates of the inscribed circle: I[11; 6.34325997374]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 136.7522014625° = 136°45'7″ = 0.75548197396 rad
∠ B' = β' = 120.0654503889° = 120°3'52″ = 1.0466071746 rad
∠ C' = γ' = 103.1833481486° = 103°11'1″ = 1.3410701168 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 = 27 ; ;

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

p = a+b+c = 19+24+27 = 70 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 70 }{ 2 } = 35 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 35 * (35-19)(35-24)(35-27) } ; ; T = sqrt{ 49280 } = 221.99 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 221.99 }{ 19 } = 23.37 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 221.99 }{ 24 } = 18.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 221.99 }{ 27 } = 16.44 ; ;

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**2-27**2 }{ 2 * 24 * 27 } ) = 43° 14'53" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-19**2-27**2 }{ 2 * 19 * 27 } ) = 59° 56'8" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-19**2-24**2 }{ 2 * 24 * 19 } ) = 76° 48'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 221.99 }{ 35 } = 6.34 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 19 }{ 2 * sin 43° 14'53" } = 13.87 ; ;




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