Video: AQA GCSE Mathematics Higher Tier Pack 5 • Paper 3 • Question 11

The area, 𝐴, of a regular hexagon with side lengths 𝑠, can be calculated using the formula 𝐴 = ((3√3)/2) 𝑠². Two regular hexagons are cut out from a piece of rectangular card, as shown in the diagram. Calculate the percentage of card remaining after the hexagons are cut out.

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Video Transcript

The area, 𝐴, of a regular hexagon with side lengths 𝑠 can be calculated using the formula 𝐴 equals three times the square root of three over two times side squared. Two regular hexagons are cut out from a piece of rectangular card, as shown in the diagram. Calculate the percentage of card remaining after the hexagons are cut out.

To find the percentage remaining, we need to consider the remaining area over the original area. But the fraction of the remaining area over the original area is not a percent. To turn it into a percentage, we need to multiply this value by 100.

Let’s think about what the remaining area would be. We need to take the area of the rectangular card we started with. To find that area, we need to multiply length times width, 50 times 65. And we find the area of the original card is 3250 centimetres squared.

Two regular hexagons were cut out from this card. And that means we need to take away, subtract, two times the area of the hexagon. We’re given a formula to find the area of a regular hexagon. The area of one of them is equal to three times the square root of three over two times side squared. But we have two regular hexagons with the same side length. And that means we need to multiply this formula by two.

When we do that, the multiplied by two and divided by two cancel. And we find the area of both of these hexagons together equals three times the square root of three times 15 squared, which equals 1169.134295 continuing. Since it’s area, we’re dealing with centimetres squared. And we’ll keep the area of the hexagons in this format so that we can round in the last step.

Back to our goal of finding the remaining area, we take the rectangular area, 3250, and we subtract the area of the two hexagons, 1169.134295 continuing. By taking out the area of the two hexagons, we find the remaining area of the rectangular card to be 2080.865705 continuing.

Now that we have the remaining area, we have all the information we need to find the percentage remaining. We plug in the remaining area, 2080.865705 continuing. We write that as a fraction over the original area of our card, 3250. And then we multiply that value by 100. 2080.865705 continuing divided by 3250 equals 0.640266 continuing, which we multiply by 100, which equals 64.0266 continuing percent.

Let’s round to the nearest percent. We look to the right of the whole number, to the deciding digit, which in this case is a zero. Zero is less than five. We’ll round down. We can round this value to 64 percent. This means that, after the two hexagons were cut out, 64 percent of the card was remaining.

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