Corners are cut off from an equilateral triangle $T$ to produce a regular hexagon $H$. Then, the ratio of the area of $H$ to the area of $T$ is

When we cut corners from an equilateral triangle to produce a regular hexagon, we're making strategic cuts that remove three corner pieces while maintaining the hexagon's regularity.

The cuts must be made parallel to the sides of the triangle to ensure the resulting shape is a regular hexagon.

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The equilateral triangle can be divided into 9 equal smaller triangles of the same size and shape.

We draw lines parallel to each side of the triangle. These lines create a grid pattern inside the triangle, where each small triangle has the same area.

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When we cut the corners to form the regular hexagon, we remove exactly 3 corner triangles (one from each corner).

The hexagon consists of 6 equal triangles from our original 9-triangle division.

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Since both the triangle and hexagon are made up of the same-sized unit triangles:

Area of Triangle T = 9 unit triangles

Area of Hexagon H = 6 unit triangles

Ratio = $\dfrac{\text{Area of H}}{\text{Area of T}}$

$= \dfrac{6}{9}$

$= \dfrac{2}{3}$

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When we cut corners from an equilateral triangle to form a regular hexagon, we're removing exactly $\tfrac{1}{3}$ of the triangle's area. This leaves us with $\tfrac{2}{3}$ of the original area.

If we imagine the triangle divided into 9 equal pieces, the 3 corner pieces that get removed are exactly those needed to transform the triangle into a regular hexagon while maintaining symmetry.

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The ratio of the area of hexagon H to the area of triangle T is $\tfrac{2}{3}$.

This makes intuitive sense because we're removing material from the triangle, so the hexagon should have a smaller area than the original triangle.