24 27 27 triangle

Acute isosceles triangle.

Sides: a = 24   b = 27   c = 27

Area: T = 290.2411278939
Perimeter: p = 78
Semiperimeter: s = 39

Angle ∠ A = α = 52.77655999225° = 52°46'32″ = 0.92111079834 rad
Angle ∠ B = β = 63.61222000388° = 63°36'44″ = 1.11102423351 rad
Angle ∠ C = γ = 63.61222000388° = 63°36'44″ = 1.11102423351 rad

Height: ha = 24.18767732449
Height: hb = 21.49993539955
Height: hc = 21.49993539955

Median: ma = 24.18767732449
Median: mb = 21.68552484422
Median: mc = 21.68552484422

Inradius: r = 7.44220840754
Circumradius: R = 15.07702202526

Vertex coordinates: A[27; 0] B[0; 0] C[10.66766666667; 21.49993539955]
Centroid: CG[12.55655555556; 7.16664513318]
Coordinates of the circumscribed circle: U[13.5; 6.69878756678]
Coordinates of the inscribed circle: I[12; 7.44220840754]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 127.2244400078° = 127°13'28″ = 0.92111079834 rad
∠ B' = β' = 116.3887799961° = 116°23'16″ = 1.11102423351 rad
∠ C' = γ' = 116.3887799961° = 116°23'16″ = 1.11102423351 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 = 24 ; ; b = 27 ; ; c = 27 ; ;

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

p = a+b+c = 24+27+27 = 78 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 78 }{ 2 } = 39 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 39 * (39-24)(39-27)(39-27) } ; ; T = sqrt{ 84240 } = 290.24 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 290.24 }{ 24 } = 24.19 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 290.24 }{ 27 } = 21.5 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 290.24 }{ 27 } = 21.5 ; ;

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{ 24**2-27**2-27**2 }{ 2 * 27 * 27 } ) = 52° 46'32" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-24**2-27**2 }{ 2 * 24 * 27 } ) = 63° 36'44" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-24**2-27**2 }{ 2 * 27 * 24 } ) = 63° 36'44" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 290.24 }{ 39 } = 7.44 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 24 }{ 2 * sin 52° 46'32" } = 15.07 ; ;




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