Video: Pack 2 β€’ Paper 2 β€’ Question 5

Pack 2 β€’ Paper 2 β€’ Question 5

02:54

Video Transcript

In the diagram, we have a parallelogram 𝐴𝐡𝐢𝐷 with a sector centred on 𝐴. The length of the base of the parallelogram is 12 centimetres and its perpendicular height is eight centimetres. The radius of the sector is four centimetres and its angle is 60 degrees. Find the area of the shaded region. Give your answer to three significant figures.

In order to find the area of the shaded region, we’re going to need to perform two separate calculations. First, we’ll need to find the area of the parallelogram, which forms the majority of the shaded area. Next, we’ll need to find the area of the sector and subtract it from the area of the parallelogram.

The formula for area of a parallelogram is the length of the base multiplied by its perpendicular height. In this case, the length of the base is 12 centimetres and the perpendicular height of the parallelogram is eight centimetres. 12 multiplied by eight is 96. The area of the parallelogram is 96 centimetres squared.

Next, we need to find the area of the sector. Sector area is given by πœƒ divided by 360 multiplied by πœ‹π‘Ÿ squared, where πœƒ is the given angle. This sector has a radius of four centimetres and an angle of 60 degrees. So this formula becomes 60 divided by 360 multiplied by πœ‹ multiplied by four squared.

Now this sort of question will usually be on a calculator paper. So we can put these numbers into the calculator and save ourselves some time. However, it is useful to know how to simplify this expression just in case you’re asked to give your answer in terms of πœ‹.

First, notice that both 60 and 360 have a highest common factor of 60. 60 divided by 60 is one and 360 divided by 60 is six. This becomes a sixth multiplied by πœ‹ multiplied by 16. Next, we notice that 16 and six have a common factor of two. 16 divided by two is eight and six divided by two is three. The area of the sector, therefore, becomes eight πœ‹ over three centimetres squared.

Remember, we said that, to find the shaded area, we would take the area of the parallelogram and subtract the area of the sector. The shaded region is, therefore, 96 minus eight πœ‹ over three. That’s 87.622, which, correct to three significant figures, is 87.6 centimetres squared.

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