Question Video: Solving Word Problems Involving the Area of Circles and Circular Sectors | Nagwa Question Video: Solving Word Problems Involving the Area of Circles and Circular Sectors | Nagwa

Question Video: Solving Word Problems Involving the Area of Circles and Circular Sectors Mathematics • First Year of Secondary School

A landscape gardener decides that he wants to design a lawn split into a series of sectors with circular patios laid into the grass, as shown in the given figure. The circular lawn will be split into six equal sectors, each with a radius of eight meters. The lines 𝑂𝐴 and 𝑂𝐵 are both tangent to the circle and the arc 𝐴𝐵 touches the circle at a single point. Work out the area of sector 𝑂𝐴𝐵. Give your answer in terms of 𝜋. The gardener needs to calculate the radius of the circular patio. Using trigonometric ratios, calculate the radius of the patio. Give your answer as a fraction. Calculate the total area of grass in one sector. Give your answer in terms of 𝜋 in its simplest form.

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Video Transcript

A landscape gardener decides that he wants to design a lawn split into a series of sectors with circular patios laid into the grass, as shown in the given figure. The circular lawn will be split into six equal sectors, each with a radius of eight meters. The lines 𝑂𝐴 and 𝑂𝐵 are both tangent to the circle and the arc 𝐴𝐵 touches the circle at a single point.

This question is broken into three parts. Let’s consider the first part.

Work out the area of sector 𝑂𝐴𝐵. Give your answer in terms of 𝜋.

Recall that to find the area of a sector, we use the formula one-half times the radius squared times 𝜃, where 𝜃 is the angle of the sector given in radians. We’ll need to convert 60 degrees to radians. And we can do that by multiplying 60 by 𝜋 over 180. 60 over 180 reduces to one over three, which makes 60 degrees 𝜋 over three radians. Now, we can plug in what we know into our formula, and we’ll have one-half times eight squared times 𝜋 over three. Eight squared is 64. 64 times one-half is 32. The area of the sector is then 32𝜋 over three. Remember, our radius was given to us in meters. And that makes the units for the area square meters.

For part two, the gardener needs to calculate the radius of the circular patio. Using trigonometric ratios, calculate the radius of the patio. Give your answer as a fraction.

There are two triangles we’ll need to consider to find the radius of the circular patio. Remember that the angle between a tangent to a circle and that circle’s radius is 𝜋 over two radians, or 90 degrees. If we let the center of the circle be point 𝑃, we can sketch a right-angle triangle whose hypotenuse runs from 𝑂 to 𝑃. If we extend our line 𝑂𝑃 up further to point 𝐶, we can say that the line 𝑂𝐶 must be equal to eight meters since it is a line that’s drawn from the center to a point on the circumference of the lawn. The line segment 𝑃𝐶 would also be a radius of the inner circle. We can say then that the line segment 𝑂𝑃 plus the line segment 𝑃𝐶 must equal eight. And the line segment 𝑃𝐶 is a radius of that circular patio. Solving for 𝑟 here, we find that the radius is equal to eight minus 𝑂𝑃.

To solve the value of the radius, we’ll need to find a second equation that expresses 𝑟 in terms of 𝑂𝑃. We can use this smaller triangle to do that. Here is 𝑂𝑃 and here’s the 𝑟 we want to write an equation for. We have an opposite side length and hypotenuse. Therefore, we’ll use the sine ratio. Remember, when we had 60 degrees, we were dealing with 𝜋 over three. We’ve cut that angle in half, which means we’re dealing with an angle of 𝜋 over six. And we can say that the sin of 𝜋 over six is equal to 𝑟 over 𝑂𝑃. sin of 𝜋 over six equals one-half. If we multiply both sides of the equation by two and then multiply both sides of the equation by 𝑂𝑃, we find that 𝑂𝑃 equals two 𝑟.

In our first equation, we can substitute two 𝑟 in place of 𝑂𝑃. If we add two 𝑟 to both sides, we get that three 𝑟 equals eight. Therefore, the radius is eight-thirds. And remember, we’re measuring in meters. So, we can say the radius of the patio will be eight-thirds meters.

For the final part of this question, calculate the total area of grass in one sector. Give your answer in terms of 𝜋 in its simplest form.

To find the area of grass, we need to take the area of the whole sector and subtract the portion that is covered with patio. We found the area of sector 𝑂𝐴𝐵 in part one to be 32𝜋 over three. The area of the patio will be equal to 𝜋𝑟 squared as it is a circle. The radius of the circular patio we found in part two to be eight-thirds. Eight-thirds squared is 64 over nine, which makes the area of the patio 64𝜋 over nine. 32𝜋 over three minus 64𝜋 over nine is equal to 32𝜋 over nine, which makes the total area covered in grass 32𝜋 over nine square meters.

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