6 27 27 triangle

Acute isosceles triangle.

Sides: a = 6   b = 27   c = 27

Area: T = 80.498844719
Perimeter: p = 60
Semiperimeter: s = 30

Angle ∠ A = α = 12.75987404169° = 12°45'31″ = 0.22326820287 rad
Angle ∠ B = β = 83.62106297916° = 83°37'14″ = 1.45994553125 rad
Angle ∠ C = γ = 83.62106297916° = 83°37'14″ = 1.45994553125 rad

Height: ha = 26.833281573
Height: hb = 5.963284794
Height: hc = 5.963284794

Median: ma = 26.833281573
Median: mb = 14.15109716981
Median: mc = 14.15109716981

Inradius: r = 2.6833281573
Circumradius: R = 13.58441129633

Vertex coordinates: A[27; 0] B[0; 0] C[0.66766666667; 5.963284794]
Centroid: CG[9.22222222222; 1.988761598]
Coordinates of the circumscribed circle: U[13.5; 1.50993458848]
Coordinates of the inscribed circle: I[3; 2.6833281573]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 167.2411259583° = 167°14'29″ = 0.22326820287 rad
∠ B' = β' = 96.37993702084° = 96°22'46″ = 1.45994553125 rad
∠ C' = γ' = 96.37993702084° = 96°22'46″ = 1.45994553125 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 = 6 ; ; b = 27 ; ; c = 27 ; ;

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

p = a+b+c = 6+27+27 = 60 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 60 }{ 2 } = 30 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 30 * (30-6)(30-27)(30-27) } ; ; T = sqrt{ 6480 } = 80.5 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 80.5 }{ 6 } = 26.83 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 80.5 }{ 27 } = 5.96 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 80.5 }{ 27 } = 5.96 ; ;

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{ 6**2-27**2-27**2 }{ 2 * 27 * 27 } ) = 12° 45'31" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-6**2-27**2 }{ 2 * 6 * 27 } ) = 83° 37'14" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-6**2-27**2 }{ 2 * 27 * 6 } ) = 83° 37'14" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 80.5 }{ 30 } = 2.68 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 6 }{ 2 * sin 12° 45'31" } = 13.58 ; ;




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