23 24 27 triangle

Acute scalene triangle.

Sides: a = 23   b = 24   c = 27

Area: T = 259.5499518304
Perimeter: p = 74
Semiperimeter: s = 37

Angle ∠ A = α = 53.21884706925° = 53°13'6″ = 0.92988375365 rad
Angle ∠ B = β = 56.69437263788° = 56°41'37″ = 0.98994921905 rad
Angle ∠ C = γ = 70.08878029287° = 70°5'16″ = 1.22332629266 rad

Height: ha = 22.56551755047
Height: hb = 21.62549598587
Height: hc = 19.2222186541

Median: ma = 22.80989894559
Median: mb = 22.02327155455
Median: mc = 19.24218814049

Inradius: r = 7.01435004947
Circumradius: R = 14.35884081556

Vertex coordinates: A[27; 0] B[0; 0] C[12.63296296296; 19.2222186541]
Centroid: CG[13.21098765432; 6.40773955137]
Coordinates of the circumscribed circle: U[13.5; 4.89901824878]
Coordinates of the inscribed circle: I[13; 7.01435004947]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 126.7821529308° = 126°46'54″ = 0.92988375365 rad
∠ B' = β' = 123.3066273621° = 123°18'23″ = 0.98994921905 rad
∠ C' = γ' = 109.9122197071° = 109°54'44″ = 1.22332629266 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 = 23 ; ; b = 24 ; ; c = 27 ; ;

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

p = a+b+c = 23+24+27 = 74 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 74 }{ 2 } = 37 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 37 * (37-23)(37-24)(37-27) } ; ; T = sqrt{ 67340 } = 259.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 259.5 }{ 23 } = 22.57 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 259.5 }{ 24 } = 21.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 259.5 }{ 27 } = 19.22 ; ;

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{ 23**2-24**2-27**2 }{ 2 * 24 * 27 } ) = 53° 13'6" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 24**2-23**2-27**2 }{ 2 * 23 * 27 } ) = 56° 41'37" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-23**2-24**2 }{ 2 * 24 * 23 } ) = 70° 5'16" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 259.5 }{ 37 } = 7.01 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 23 }{ 2 * sin 53° 13'6" } = 14.36 ; ;




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