3 27 27 triangle

Acute isosceles triangle.

Sides: a = 3   b = 27   c = 27

Area: T = 40.43774517001
Perimeter: p = 57
Semiperimeter: s = 28.5

Angle ∠ A = α = 6.36994770734° = 6°22'10″ = 0.11111683466 rad
Angle ∠ B = β = 86.81552614633° = 86°48'55″ = 1.51552121535 rad
Angle ∠ C = γ = 86.81552614633° = 86°48'55″ = 1.51552121535 rad

Height: ha = 26.95883011334
Height: hb = 2.99553667926
Height: hc = 2.99553667926

Median: ma = 26.95883011334
Median: mb = 13.66656503687
Median: mc = 13.66656503687

Inradius: r = 1.41988579544
Circumradius: R = 13.5210881683

Vertex coordinates: A[27; 0] B[0; 0] C[0.16766666667; 2.99553667926]
Centroid: CG[9.05655555556; 0.99884555975]
Coordinates of the circumscribed circle: U[13.5; 0.75111600935]
Coordinates of the inscribed circle: I[1.5; 1.41988579544]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 173.6310522927° = 173°37'50″ = 0.11111683466 rad
∠ B' = β' = 93.18547385367° = 93°11'5″ = 1.51552121535 rad
∠ C' = γ' = 93.18547385367° = 93°11'5″ = 1.51552121535 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 = 3 ; ; b = 27 ; ; c = 27 ; ;

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

p = a+b+c = 3+27+27 = 57 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 57 }{ 2 } = 28.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 28.5 * (28.5-3)(28.5-27)(28.5-27) } ; ; T = sqrt{ 1635.19 } = 40.44 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 40.44 }{ 3 } = 26.96 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 40.44 }{ 27 } = 3 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 40.44 }{ 27 } = 3 ; ;

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{ 3**2-27**2-27**2 }{ 2 * 27 * 27 } ) = 6° 22'10" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-3**2-27**2 }{ 2 * 3 * 27 } ) = 86° 48'55" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-3**2-27**2 }{ 2 * 27 * 3 } ) = 86° 48'55" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 40.44 }{ 28.5 } = 1.42 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 3 }{ 2 * sin 6° 22'10" } = 13.52 ; ;




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