Minimizing Surface Area: find maximum area of hexagon inscribed in circle

[yli]

Hi, I needed some help with this problem
Determine the ratio h/r that minimizes the "effective surface area'' A if we assume that the top and bottom of the can are cut from hexagons of sheet metal.

I have been trying to determine the maximum amount of area of a hexagon, inscribed in a circle, but I've gotten somewhat confused about it. I know that a hexagon inscribed in a circle is made up of 6 equilateral triangles, and so I have been trying to find a formula for the area using that fact, but I've gotten stuck. I was thinking that all the sides of the triangles would be equal to the radius, and therefore because the area of a triangle is bh/2, I was thinking it could be something along the lines of 2r/2. Any help is appreciated.

 

[mmm4444bot]

… hexagon [area] is made up of 6 equilateral [triangle areas], and so I have been trying to find a formula for the area using that fact, but I've gotten stuck. I was thinking that all the sides of the triangles would be equal to the radius, and therefore because the area of a triangle is bh/2, I was thinking it could be something along the lines of 2r/2. Any help is appreciated.

The height of each triangle is not r because the base of each triangle is inside the circle.

If you were to draw a line from the hexagon's center to the midpoint of each base, you would create 12 congruent right triangles. Now you can find the height of each, in terms of r, using the Pythagorean Theorem. :cool:

 

[Harry_the_cat]

Hi, I needed some help with this problem
Determine the ratio h/r that minimizes the "effective surface area'' A if we assume that the top and bottom of the can are cut from hexagons of sheet metal.

I have been trying to determine the maximum amount of area of a hexagon, inscribed in a circle, but I've gotten somewhat confused about it. I know that a hexagon inscribed in a circle is made up of 6 equilateral triangles, and so I have been trying to find a formula for the area using that fact, but I've gotten stuck. I was thinking that all the sides of the triangles would be equal to the radius, and therefore because the area of a triangle is bh/2, I was thinking it could be something along the lines of 2r/2. Any help is appreciated.

You could use the trig formula for the area of the triangle Area =0.5ab sin C where a=r, b=r and C=60 degrees.

 

[stapel]

Determine the ratio h/r that minimizes the "effective surface area'' A if we assume that the top and bottom of the can are cut from hexagons of sheet metal.

Your subject line refers to "minimizing surface area". "Surface area" refers to the area of the surfaces of a three-dimensional object. You have describing maximizing the plain old "area" of a two-dimensional object. Is there more information for this exercise?

I have been trying to determine the maximum amount of area of a hexagon, inscribed in a circle, but I've gotten somewhat confused about it. I know that a hexagon inscribed in a circle is made up of 6 equilateral triangles...

Not necessarily, though the maximizing hexagon will have this shape. (It's like inscribing triangles. Yes, the triangle with the maximal area will be equilateral, but there are infinitely-many other triangles which can be inscribed.) Are you allowed to assume this shape, or is that kinda what you're supposed to be proving?

...so I have been trying to find a formula for the area using that fact, but I've gotten stuck. I was thinking that all the sides of the triangles would be equal to the radius, and therefore because the area of a triangle is bh/2, I was thinking it could be something along the lines of 2r/2. Any help is appreciated.

In this case, are you taking "h" to be "the distance from the center of the hexagon to the furthest point on a chord of the circle formed by the side of a hexagon"? Are you assuming that the hexagon's center and the circle's center are the same? Are you assuming that all of the sides of the hexagon are all the same? Does "b" represent "the length of the chord corresponding to the base of one of the hexagon's triangles"? Are you assuming them all to be the same? Does "r" represent "the radius of the circumscribed circle"?

When you reply, please include a clear listing of your efforts, up to where it got "somewhat confused". Thank you!