Video: AQA GCSE Mathematics Foundation Tier Pack 4 • Paper 3 • Question 16

A wire stencil is made using a circle and three regular hexagons, as shown. The hexagons have a side length of 7 cm. Work out the total length of the wire.

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Video Transcript

A wire stencil is made using a circle and three regular hexagons, as shown. The hexagons have a side length of seven centimeters. Work out the total length of the wire.

A hexagon has six sides. And if the hexagon is regular, then each side is the same length, in this case seven centimeters. As there are three hexagons, we would expect there to be a total of 18 sides as three multiplied by six is equal to 18. However, there are three sides that are shared between two hexagons. So they will not be 18 sides in total. In fact, we have 12 sides on the outside of the wire stencil. Adding the three sides inside the circle gives us a total of 15 sides.

The total length of the wire for the hexagons will, therefore, be seven centimeters multiplied by 15. Seven multiplied by 15 is equal to 105 as seven multiplied by 10 is 70 and seven multiplied by five is 35. The total length of the wire used for the three hexagons is 105 centimeters. In order to work out the amount of wire needed for the circle, we need to calculate its circumference. We do this using the formula 𝜋 multiplied by 𝑑 the diameter. We can see from the diagram that the side of a hexagon is the radius of the circle. This is equal to seven centimeters. If the radius is equal to seven centimeters, then the diameter must be equal to 14 centimeters as the diameter is double the length of the radius.

We can, therefore, calculate the amount of wire required to make the circle by multiplying 𝜋 by 14 centimeters. This is equal to 43.982 and so on centimeters. We can then calculate the total length of the wire by adding 105 and 43.982 et cetera. We can round this answer of 148.982 to the nearest whole number. We do this by looking at the number immediately after the decimal point. As the nine is greater than five, we will round up. The total length of wire that is required to make the stencil is 149 centimeters. This is made up of 15 sides of length seven centimeters and the circumference of a circle with diameter 14 centimeters.

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