8 27 27 triangle

Acute isosceles triangle.

Sides: a = 8   b = 27   c = 27

Area: T = 106.8088239383
Perimeter: p = 62
Semiperimeter: s = 31

Angle ∠ A = α = 17.03992485083° = 17°2'21″ = 0.29773909885 rad
Angle ∠ B = β = 81.48803757459° = 81°28'49″ = 1.42221008325 rad
Angle ∠ C = γ = 81.48803757459° = 81°28'49″ = 1.42221008325 rad

Height: ha = 26.70220598456
Height: hb = 7.91217214357
Height: hc = 7.91217214357

Median: ma = 26.70220598456
Median: mb = 14.63772811683
Median: mc = 14.63772811683

Inradius: r = 3.44554270769
Circumradius: R = 13.65106322773

Vertex coordinates: A[27; 0] B[0; 0] C[1.18551851852; 7.91217214357]
Centroid: CG[9.39550617284; 2.63772404786]
Coordinates of the circumscribed circle: U[13.5; 2.02223158929]
Coordinates of the inscribed circle: I[4; 3.44554270769]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 162.9610751492° = 162°57'39″ = 0.29773909885 rad
∠ B' = β' = 98.52196242541° = 98°31'11″ = 1.42221008325 rad
∠ C' = γ' = 98.52196242541° = 98°31'11″ = 1.42221008325 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 = 8 ; ; b = 27 ; ; c = 27 ; ;

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

p = a+b+c = 8+27+27 = 62 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 62 }{ 2 } = 31 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 31 * (31-8)(31-27)(31-27) } ; ; T = sqrt{ 11408 } = 106.81 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 106.81 }{ 8 } = 26.7 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 106.81 }{ 27 } = 7.91 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 106.81 }{ 27 } = 7.91 ; ;

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{ 8**2-27**2-27**2 }{ 2 * 27 * 27 } ) = 17° 2'21" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-8**2-27**2 }{ 2 * 8 * 27 } ) = 81° 28'49" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-8**2-27**2 }{ 2 * 27 * 8 } ) = 81° 28'49" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 106.81 }{ 31 } = 3.45 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 8 }{ 2 * sin 17° 2'21" } = 13.65 ; ;




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