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

## Join Nagwa Classes

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 ππ΄ and ππ΅ are both tangent to the circle and the arc π΄π΅ touches the circle at a single point. Work out the area of sector ππ΄π΅. 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.

04:53

### 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 ππ΄ and ππ΅ are both tangent to the circle and the arc π΄π΅ touches the circle at a single point.

This question is broken into three parts. Letβs consider the first part.

Work out the area of sector ππ΄π΅. 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 π, where π is the angle of the sector given in radians. Weβ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 weβ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 weβll need to consider to find the radius of the circular patio. Remember that the angle between a tangent to a circle and that circleβs radius is π over two radians, or 90 degrees. If we let the center of the circle be point π, we can sketch a right-angle triangle whose hypotenuse runs from π to π. If we extend our line ππ up further to point πΆ, we can say that the line ππΆ must be equal to eight meters since it is a line thatβs drawn from the center to a point on the circumference of the lawn. The line segment ππΆ would also be a radius of the inner circle. We can say then that the line segment ππ plus the line segment ππΆ must equal eight. And the line segment ππΆ is a radius of that circular patio. Solving for π here, we find that the radius is equal to eight minus ππ.

To solve the value of the radius, weβll need to find a second equation that expresses π in terms of ππ. We can use this smaller triangle to do that. Here is ππ and hereβs the π we want to write an equation for. We have an opposite side length and hypotenuse. Therefore, weβll use the sine ratio. Remember, when we had 60 degrees, we were dealing with π over three. Weβve cut that angle in half, which means weβre dealing with an angle of π over six. And we can say that the sin of π over six is equal to π over ππ. 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 ππ, we find that ππ equals two π.

In our first equation, we can substitute two π in place of ππ. If we add two π to both sides, we get that three π equals eight. Therefore, the radius is eight-thirds. And remember, weβ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 ππ΄π΅ in part one to be 32π over three. The area of the patio will be equal to ππ 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.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions