Okay, so I need to figure out the perimeter of a regular hexagon when one of the equilateral triangles formed by dividing the hexagon has a perimeter of 21 inches. Let me start by understanding the problem step by step.

First, I recall that a regular hexagon can indeed be divided into six equilateral triangles. Each of these triangles has two sides that are radii of the hexagon and one side that is a side of the hexagon itself. Wait, actually, when you divide a regular hexagon into triangles by drawing lines from the center to each vertex, each triangle is equilateral. So all sides of each triangle are equal. That means each triangle's sides are equal to the radius of the hexagon, right?

But hold on, in a regular hexagon, the radius (distance from the center to a vertex) is equal to the length of each side of the hexagon. Because in a regular hexagon, all sides are equal, and the distance from the center to any vertex is the same as the side length. So if each triangle is equilateral, then all three sides of the triangle are equal to the side length of the hexagon. That makes sense.

So if the perimeter of one of these triangles is 21 inches, then since all three sides are equal, each side must be 21 divided by 3, which is 7 inches. Therefore, each side of the hexagon is 7 inches. Since a hexagon has six sides, the perimeter would be 6 times 7 inches. Let me check that again.

Wait, perimeter of the triangle is 21 inches. If the triangle is equilateral, each side is 21/3 = 7 inches. Since the side of the triangle is the same as the side of the hexagon, the hexagon's side length is 7 inches. Therefore, the perimeter of the hexagon is 6 * 7 = 42 inches. That seems straightforward. But let me make sure I didn't skip any steps or make any incorrect assumptions.

Let me visualize a regular hexagon. If you connect the center to each vertex, you create six equilateral triangles. Each triangle has two sides that are radii (which are equal to the side length of the hexagon) and one side that's a side of the hexagon. Therefore, each triangle is equilateral. So yes, each side of the triangle is equal to the side length of the hexagon. Therefore, the side length of the hexagon is 7 inches, so perimeter is 6*7=42 inches. That seems correct.

But let me think again. Suppose the side length of the triangle is not the same as the side length of the hexagon. Wait, in a regular hexagon, the radius (distance from center to vertex) is equal to the side length. So when you split the hexagon into triangles, each triangle has two sides equal to the radius (which is the side length of the hexagon) and one side equal to the side length of the hexagon. Therefore, all three sides of the triangle are equal, making it equilateral. So yes, the side length of the triangle is equal to the side length of the hexagon.

Therefore, if the triangle's perimeter is 21 inches, each side is 7 inches. Therefore, the hexagon's side is 7 inches, so perimeter is 6*7=42 inches.

Hmm, seems straightforward. Maybe I'm overcomplicating it, but I want to make sure there's no trick here. Let me check with another approach.

Alternatively, the perimeter of the triangle is 21 inches. Since the triangle is equilateral, all sides are equal. So side length of triangle is 21/3=7 inches. Since the side of the triangle is the same as the side of the hexagon (as each side of the hexagon is the base of one of these triangles), then the hexagon's side is 7. Therefore, perimeter is 6*7=42. Yep, same answer.

Alternatively, maybe there's a formula I can use. For a regular hexagon, the perimeter is 6 times the side length. If the hexagon is divided into six equilateral triangles, each triangle has a side equal to the hexagon's side. So the perimeter of the triangle is 3 times the side of the hexagon. Therefore, perimeter of hexagon is 6 times the side, which is 2 times the perimeter of one triangle. Wait, 3 times side = triangle perimeter, so side = triangle perimeter /3. Then hexagon perimeter is 6*(triangle perimeter /3) = 2*triangle perimeter. So 2*21=42. That's another way to look at it. So this confirms the answer is 42 inches.

Yes, that makes sense. So regardless of the method, the perimeter comes out to be 42 inches. I think that's solid. I can't see any flaws in this reasoning. So I'm confident the answer is 42.
Therefore, the perimeter of the regular hexagon is $\boxed{42}$ inches.
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