22 24 27 triangle

Acute scalene triangle.

Sides: a = 22   b = 24   c = 27

Area: T = 250.6965906428
Perimeter: p = 73
Semiperimeter: s = 36.5

Angle ∠ A = α = 50.69220933471° = 50°41'32″ = 0.88547439336 rad
Angle ∠ B = β = 57.57549908583° = 57°34'30″ = 1.00548731573 rad
Angle ∠ C = γ = 71.73329157946° = 71°43'59″ = 1.25219755627 rad

Height: ha = 22.7910536948
Height: hb = 20.89113255357
Height: hc = 18.57700671429

Median: ma = 23.05442837668
Median: mb = 21.50658131676
Median: mc = 18.6488056199

Inradius: r = 6.8688380998
Circumradius: R = 14.21664267888

Vertex coordinates: A[27; 0] B[0; 0] C[11.79662962963; 18.57700671429]
Centroid: CG[12.93220987654; 6.1990022381]
Coordinates of the circumscribed circle: U[13.5; 4.45660958969]
Coordinates of the inscribed circle: I[12.5; 6.8688380998]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 129.3087906653° = 129°18'28″ = 0.88547439336 rad
∠ B' = β' = 122.4255009142° = 122°25'30″ = 1.00548731573 rad
∠ C' = γ' = 108.2677084205° = 108°16'1″ = 1.25219755627 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 = 22 ; ; b = 24 ; ; c = 27 ; ;

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

p = a+b+c = 22+24+27 = 73 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 73 }{ 2 } = 36.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 36.5 * (36.5-22)(36.5-24)(36.5-27) } ; ; T = sqrt{ 62848.44 } = 250.7 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 250.7 }{ 22 } = 22.79 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 250.7 }{ 24 } = 20.89 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 250.7 }{ 27 } = 18.57 ; ;

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{ 22**2-24**2-27**2 }{ 2 * 24 * 27 } ) = 50° 41'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-22**2-27**2 }{ 2 * 22 * 27 } ) = 57° 34'30" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-22**2-24**2 }{ 2 * 24 * 22 } ) = 71° 43'59" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 250.7 }{ 36.5 } = 6.87 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 22 }{ 2 * sin 50° 41'32" } = 14.22 ; ;




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