help hexagon geometry

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.

  1. 👍
  2. 👎
  3. 👁
  1. 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

    1. 👍
    2. 👎
  2. iiitss hahaha nto telling you

    1. 👍
    2. 👎
  3. drwls you were close it's 4/3

    1. 👍
    2. 👎
  4. yo i need help

    1. 👍
    2. 👎
  5. 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]

    1. 👍
    2. 👎
  6. 4/3 is the answer.

    1. 👍
    2. 👎
  7. It is 4/3.

    1. 👍
    2. 👎

Respond to this Question

First Name

Your Response

Similar Questions

  1. Math please helppp

    1 Find the missing angle measure in the polygon A ] 77 B ] 87 C ] 97 D ] 107 i think its b 2 Find the sum of the interior angles in 10 - sided polygon A ] 1,260 B ] 1,440 c ] 1,620 d ] 1,800 i think its c or b 3 Find the measure

  2. geometry please please please help

    What is the value of x in the regular polygon below? hexagon with an interior angle value of 3x please please help and thanks in advance. :)

  3. Maths

    If ABCDEF is a regular hexagon then prove that ab+ac+ad+ea+fa=4ab

  4. easy geometry

    1)If diagonals of a rhombus are 10 cm and 24 cm. find the area and perimeter of the rhombus. 2)A regular hexagon with a perimeter of 24 units is inscribed in a circle. Find the radius of the circle. 3)Find the altitude,perimeter

  1. Solid Mensuration

    The area of a regular hexagon inscribed in a circle is equal to 166.28 square cm. If the circle is inscribed in a square, find the difference between the area of the square and the hexagon.

  2. Geometry

    Hi, I'm having a difficult time understanding a couple of Geometry questions on my homework and I'd really appreciate it if someone could answer them and explain to me how to get the same answer. Question 1: A regular pentagon has

  3. math

    Two regular hexagons are similar. The perimeters have a ratio of 5: 2. If the side of the larger hexagon has a length of 25 inches, what is the perimeter of the smaller hexagon? AND COULD SOMEONE PLEASE ANSWER IT!?

  4. Geometry

    What Is The Measure,In Degrees, Of Each Exterior Angle Of A Regular Hexagon ?

  1. Geometry / Trig

    What is the area of a regular hexagon with a distance from its center to a vertex of 1 cm? (Hint: A regular hexagon can be divided into six equilateral triangles.) hex The area of the regular hexagon is _____ ?

  2. Math

    I have two questions that I need help with. 1. Angle of Depression: A Global Positioning System satellite orbits 12,500 miles above Earth's surface. Find the angle of depression from the satellite to the horizon. Assume the radius

  3. Solid Geometry

    Find the Lateral area in cm square of the following right pryramid. 1. Base is regular octagon of side 20cm and altitude of 20cm. 2. Base is regular hexagon of side 20cm and altitude of 30cm. 3. Base is regular hexagon of side

  4. Maths

    A regular hexagon is to be cut from a sheet of diameter d.the width across corner is

You can view more similar questions or ask a new question.