13 27 27 triangle

Acute isosceles triangle.

Sides: a = 13   b = 27   c = 27

Area: T = 170.3388449858
Perimeter: p = 67
Semiperimeter: s = 33.5

Angle ∠ A = α = 27.86105271851° = 27°51'38″ = 0.48662579307 rad
Angle ∠ B = β = 76.07697364075° = 76°4'11″ = 1.32876673614 rad
Angle ∠ C = γ = 76.07697364075° = 76°4'11″ = 1.32876673614 rad

Height: ha = 26.20659153628
Height: hb = 12.61876629524
Height: hc = 12.61876629524

Median: ma = 26.20659153628
Median: mb = 16.33224829711
Median: mc = 16.33224829711

Inradius: r = 5.08547298465
Circumradius: R = 13.90990733888

Vertex coordinates: A[27; 0] B[0; 0] C[3.13296296296; 12.61876629524]
Centroid: CG[10.04332098765; 4.20658876508]
Coordinates of the circumscribed circle: U[13.5; 3.34884806306]
Coordinates of the inscribed circle: I[6.5; 5.08547298465]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 152.1399472815° = 152°8'22″ = 0.48662579307 rad
∠ B' = β' = 103.9330263593° = 103°55'49″ = 1.32876673614 rad
∠ C' = γ' = 103.9330263593° = 103°55'49″ = 1.32876673614 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 = 13 ; ; b = 27 ; ; c = 27 ; ;

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

p = a+b+c = 13+27+27 = 67 ; ;

2. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 67 }{ 2 } = 33.5 ; ;

3. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 33.5 * (33.5-13)(33.5-27)(33.5-27) } ; ; T = sqrt{ 29015.19 } = 170.34 ; ;

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

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 170.34 }{ 13 } = 26.21 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 170.34 }{ 27 } = 12.62 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 170.34 }{ 27 } = 12.62 ; ;

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{ 13**2-27**2-27**2 }{ 2 * 27 * 27 } ) = 27° 51'38" ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 27**2-13**2-27**2 }{ 2 * 13 * 27 } ) = 76° 4'11" ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 27**2-13**2-27**2 }{ 2 * 27 * 13 } ) = 76° 4'11" ; ;

6. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 170.34 }{ 33.5 } = 5.08 ; ;

7. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 13 }{ 2 * sin 27° 51'38" } = 13.91 ; ;




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