help hexagon geometry

posted by .

A regular hexagon is inscribed in a circle and another regular hexagon is circumscribed about the same circle. What is the ratio of the area of the larger hexagon to the area of the smaller hexagon? Express your answer as a common fraction.

How is this done? Would we divide it into triangles?
Thanks a lot for helping.

  • help hexagon geometry -

    Realize that a hexagon is six equilateral triangles stacked together. You need the radii if the inscribed and circumscribed circles. The area ratio is the square of the ratio of those two radii.

    The circumscribed circle radius is the side length of the equilateral triangles. Call that a. The inscribed circle has a radius that is the height of the equilateral triangles, a*sqrt3/2

    The square of the radius ratio is 3/4

  • help hexagon geometry -

    That's not the answer hes dumb

  • help hexagon geometry -

    iiitss hahaha nto telling you

  • help hexagon geometry -

    drwls you were close it's 4/3

  • help hexagon geometry -

    yo i need help

  • help hexagon geometry -

    Form a triangle whose first vertex is the center of the circle and whose other two vertices are the midpoint and one of the endpoints of a side of the larger hexagon, as shown in the diagram. Since each interior angle of a regular hexagon is 120 degrees, this triangle is a 30-60-90 right triangle. Let $r$ be the radius of the circle. The length of the longer leg of the triangle is $r$, so the length of the shorter leg is $r/\sqrt{3}$ and the length of the hypotenuse is $2r/\sqrt{3}$. Since for the smaller hexagon the length of the segment connecting a vertex to the center is $r$, the dimensions of the larger hexagon are $2/\sqrt{3}$ times larger than the dimensions of the smaller hexagon. Therefore, the area of the larger triangle is $(2/\sqrt{3})^2=\boxed{4/3}$ times greater than the area of the smaller triangle.

    [asy]
    size(5cm);
    defaultpen(linewidth(.7pt)+fontsize(8pt));
    dotfactor=4;
    int i;
    draw(circle((0,0),1));
    for(i=0;i<=5;++i)

    {

    draw(dir(60*i)--dir(60*(i+1)));

    draw(2/sqrt(3)*dir(60*i)--2/sqrt(3)*dir(60*(i+1)));

    }
    draw(2/sqrt(3)*dir(0)--(0,0)--dir(30));
    draw(0.93*dir(30)--dir(30)+0.07*dir(-60)+0.07*dir(210)--dir(30)+0.07*dir(-60));[/asy]

  • help hexagon geometry -

    4/3 is the answer.

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. math

    how do you find the area of a hexagon? Here is the formula: Area of regular hexagon: (sqrt3/ 2)(Wsqrd) Where sqrt=square root W=the smallest width of the hexagon sqrd=squared
  2. Geometry

    A regular hexagon with sides of 3" is inscribed in a circle. What is the area of a segment formed by a side of the hexagon and the circle?
  3. geometry

    a regular hexagon is inscribed in a circle. The radius of the circle is 18 units. What is the area of the region bounded by the inside of the circle and the outside of the hexagon. Round your answer to the nearest hundredth.
  4. geometry

    The area of a particular regular hexagon is x^3 square units, where x is the measure of the distance from the center of the hexagon to the midpoint of a side. what is the side length of a hexagon?
  5. geometry

    Hexagon D and hexagon H are regular hexagons. The scale factor from hexagon D to hexagon H is 0.25. One side of hexagon D measures 18 cm. What is the length of one side of hexagon H?
  6. geometry

    Hexagon A is a regular hexagon. The total length of all the sides of the hexagon is 24 inches. Hexagon A is dilated about its center to create Hexagon B. The length of each side of Hexagon B is inches. By what factor was Hexagon A …
  7. math

    A regular hexagon with sides of 3" is inscribed in a circle. Find the area of a segment formed by a side of the hexagon and the circle
  8. singh

    a circle inscribed in regular hexagon with each side of regular hexagon √3 cm find area of circle
  9. Geometry / Trig

    What is the area of a regular hexagon with a distance from its center to a vertex of 1 cm?
  10. math

    the area of a regular hexagon is given as 384√3 a. how long is each side of a hexagon b. find the radius of a hexagon c. find the apothem of a hexagon

More Similar Questions