14 24 27 triangle

Acute scalene triangle.

Sides: a = 14   b = 24   c = 27

Area: T = 167.6565711206
Perimeter: p = 65
Semiperimeter: s = 32.5

Angle ∠ A = α = 31.16217524168° = 31°9'42″ = 0.54438751804 rad
Angle ∠ B = β = 62.50770018946° = 62°30'25″ = 1.09109529886 rad
Angle ∠ C = γ = 86.33112456886° = 86°19'52″ = 1.50767644846 rad

Height: ha = 23.95108158866
Height: hb = 13.97113092672
Height: hc = 12.41989415708

Median: ma = 24.56662369931
Median: mb = 17.84765682976
Median: mc = 14.27441024236

Inradius: r = 5.15986372679
Circumradius: R = 13.52877228773

Vertex coordinates: A[27; 0] B[0; 0] C[6.4632962963; 12.41989415708]
Centroid: CG[11.15443209877; 4.14396471903]
Coordinates of the circumscribed circle: U[13.5; 0.86656132198]
Coordinates of the inscribed circle: I[8.5; 5.15986372679]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 148.8388247583° = 148°50'18″ = 0.54438751804 rad
∠ B' = β' = 117.4932998105° = 117°29'35″ = 1.09109529886 rad
∠ C' = γ' = 93.66987543114° = 93°40'7″ = 1.50767644846 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 = 14 ; ; b = 24 ; ; c = 27 ; ;

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

p = a+b+c = 14+24+27 = 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 * (32.5-14)(32.5-24)(32.5-27) } ; ; T = sqrt{ 28108.44 } = 167.66 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 167.66 }{ 14 } = 23.95 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 167.66 }{ 24 } = 13.97 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 167.66 }{ 27 } = 12.42 ; ;

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{ 14**2-24**2-27**2 }{ 2 * 24 * 27 } ) = 31° 9'42" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-14**2-27**2 }{ 2 * 14 * 27 } ) = 62° 30'25" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-14**2-24**2 }{ 2 * 24 * 14 } ) = 86° 19'52" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 167.66 }{ 32.5 } = 5.16 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 14 }{ 2 * sin 31° 9'42" } = 13.53 ; ;




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