**Grade 11 Jar Question**

__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

the area

than the

We know that:

- A

- A

By equating these, we can find the **ratio of r **(the radius of the circular base)

of the hexagon). Then we can decide the relative size of 6

________________________

__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

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

- when we square root both sides,

**Step 3**:

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

- We divide both of them by 2

- ,

- 2.857

__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

This means that *r*², (the square of the circle's radius) is 25980.76211 ÷ = 8269.93...

- when we square root it, we get that

**Step 3**:

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

- We divide both of them by 2

- Since ,

285.7 < 300, so the sides of the circular jar require less material.

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