An equilateral triangle has area 300 square centimetres. From each of its three vertices, a small equilateral triangle is cut off so that the remaining central figure is a regular hexagon. The area of this regular hexagon is what percentage of the area of the original triangle?

Introduction / Context:
This geometry question explores the relationship between an equilateral triangle and a regular hexagon formed inside it by cutting off three smaller congruent equilateral triangles from the corners. The central figure is a regular hexagon, and we must find its area as a percentage of the area of the original triangle. This setup is closely related to classical constructions in which trisection points of the sides of an equilateral triangle are joined to form a regular hexagon inside the triangle.

Given Data / Assumptions:

• The large figure is an equilateral triangle of area 300 cm^2.
• From each of its three vertices, a smaller equilateral triangle is cut off.
• The remaining central figure is a regular hexagon.
• All small removed triangles are congruent.
• We must find the area of the hexagon as a percentage of the area of the original triangle.

Concept / Approach:
In the standard construction, points at one third of each side of an equilateral triangle are joined to form a regular hexagon in the middle. The three removed corner pieces are equilateral triangles whose side length is one third of the side of the original triangle, while the original triangle has side length three units in that same scale. Using the area formula for equilateral triangles, we can find the ratio of the hexagon area to the triangle area as a pure number, which applies for any size, and then convert that ratio into a percentage and apply it to the given area 300 cm^2.

Step-by-Step Solution:
Step 1: Consider an equilateral triangle with side length 3 units for simplicity. Step 2: Divide each side into three equal segments and join suitable trisection points. This construction cuts off three corner equilateral triangles, each with side length 1 unit, and leaves a regular hexagon in the centre. Step 3: Area of an equilateral triangle with side s is (√3 / 4) * s^2. Step 4: The area of the original large triangle with side 3 is A_large = (√3 / 4) * 3^2 = (9√3 / 4). Step 5: Each small cut off triangle has side 1, so its area is (√3 / 4) * 1^2 = √3 / 4. Step 6: Three such triangles are removed, so total removed area = 3 * (√3 / 4) = 3√3 / 4. Step 7: Area of the central hexagon = A_large − removed area = (9√3 / 4) − (3√3 / 4) = 6√3 / 4 = 3√3 / 2. Step 8: Ratio of hexagon area to triangle area = (3√3 / 2) / (9√3 / 4) = (3√3 / 2) * (4 / 9√3) = 12 / 18 = 2 / 3. Step 9: Therefore, the hexagon has 2 / 3 of the area of the original triangle, which is 66.66% (approximately) of the triangle area. Step 10: For a triangle of area 300 cm^2, the hexagon area is (2 / 3) * 300 = 200 cm^2, which still represents 66.66% of the original area.

Verification / Alternative check:
Because the ratio 2 / 3 was obtained purely from similarity and exact formulas, it does not depend on the actual size of the triangle. Any equilateral triangle undergoing this corner cutting construction will produce a central hexagon with area equal to two thirds of the original area. Using the numeric value 200 / 300 = 2 / 3 confirms that the percentage is 66.66% (repeating). This validates both the geometric reasoning and the arithmetic.

Why Other Options Are Wrong:
33.33% corresponds to one third of the area, which would be the removed area, not the remaining central region. 83.33% and 56.41% come from incorrect assumptions about the side ratios or from wrongly identified small triangle sizes. 50.00% would be the case if only half the area were removed or retained, but our construction shows that exactly one third of the area is cut off and two thirds remains.

Common Pitfalls:
Some students assume the side of the hexagon is half the side of the triangle rather than being related to trisection points, which changes the area ratio. Others may neglect that three congruent triangles are removed and forget to multiply by three. Another mistake is to compute numeric areas for a particular side length but then miscalculate the final percentage by mixing up removed and remaining area. Keeping the ratios symbolic until the end and clearly labelling which region is the hexagon avoids these issues.

Final Answer:
The area of the hexagon is 66.66% of the area of the triangle.