QUESTION

We have 2 glass jars, both of which contain 375 mL of jam.
The base of one is a regular hexagon, the other is a circle.
The bases have the same area.

Which jar requires less glass to make the sides if the
thickness of the glass for both jars is the same?

Overview:

Since both the volumes and the base areas are equal, the height h of both
jars must be equal
also. The solution then is simply to figure out whether
the area
of the 6 rectangles (the sides of the hexagonal jar) is greater or less
than the area of the cylinder (the sides of the circular jar).

We know that:

A(hexagon) = ½(6)(s)(a) with a = apothem.
A(circle) =

By equating these, we can find the ratio of r (the radius of the circular base) to s (the side
of the hexagon). Then we can decide the relative size of 6sh and , the lateral areas.

________________________

Solution 1 -- just variables:

Step 1:

The interior angles in a hexagon = 120°.
When we draw the apothem and the radius,
we get a 30°, 60°, 90° triangle, and we know the ratio of the sides.

If we set s = the side of the hexagon, we know the apothem, . (as shown)

Step 2:

Find the area of the hexagonal base, set it = the area of a circle with radius r.

Since we know this must be the same as , we can find r in terms of s.

when we square root both sides, r = 0.9094 s.

Step 3:

Compare the values of 6sh (the area of the 6 rectangles) with , (the area of the cylinder).

We divide both of them by 2h and compare 3s with .
,
2.857s < 3s, so the sides of the circular jar require 14.3% less material.

Solution 2 -- with numbers:

Step 1:

Since the interior angles in a hexagon = 120°,
when we draw the apothem and the radius,
we get a 30°, 60°, 90° triangle, for which we know the ratio of the sides.

If we set s the side of the hexagon = 100 units we know the apothem. (as shown)

Step 2:

Find the area of the hexagonal base, then find r, the radius of a circle of equal area.

A(hexagon) = ½(100) (6) = 25980.76211 units².

This means that r², (the square of the circle's radius) is 25980.76211 ÷ = 8269.93...

when we square root it, we get that r = 90.94 units.

Step 3:

Compare 600h (the area of the 6 rectangles) with , the lateral area of the cylinder.

We divide both of them by 2h and compare 300 with .
Since ,
285.7 < 300, so the sides of the circular jar require less material.

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