Video: Solving Problems Involving Sectors

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 yards. The lines 𝑂𝐴 and 𝑂𝐵 are both tangents 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.

Recall, the formula for the area of a sector with angle 𝜃 radians and radius 𝑟 is a half multiplied by 𝑟 squared multiplied by 𝜃. We can’t currently use this formula to work out the area of sector 𝑂𝐴𝐵 since the included angle is in degrees rather than radians. First, we’ll need to change the 60 degrees to radians by multiplying by 𝜋 over 180.

We can cross-cancel by a factor of 60. And this tells us that 60 degrees is equivalent to 𝜋 over three radians. Now let’s substitute what we know about the sector into the formula for its area. It’s a half multiplied by the radius squared. That’s eight squared multiplied by 𝜋 over three. Eight squared is 64. And a half of 64 is 32. So, a half multiplied by eight squared multiplied by 𝜋 over three is 32𝜋 over three. And we can say that the area of the sector is 32𝜋 over three square yards.

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’re actually two triangles we’re going to need to consider to help us find the radius of the circular patio. Remember, the angle between the tangent to a circle and the circle’s radius is 𝜋 over two radians. We can therefore add in the radius of the patio forming a right-angled triangle at point 𝑃, which is the centre of the circle. We also know that if we extend the line 𝑂𝑃 up to the circumference of the circle at 𝐶, that whole line is eight yards, since it’s the line joining the centre of the lawn to a point on its circumference. It’s the radius of the lawn itself.

The line segment 𝑃𝐶 is also the radius of the circle. Since 𝑂𝑃 and 𝑃𝐶 give a total length of eight yards, and 𝑃𝐶 is the radius of the circle, we can say that 𝑂𝑃 plus 𝑟 is equal to eight. We can subtract 𝑂𝑃 from both sides of this equation, and that tells us that the radius is eight minus 𝑂𝑃. Now let’s consider the small right-angled triangle. We can use right-angle trigonometry to find an expression for the radius of the patio in terms of the line 𝑂𝑃.

We can use the sine ratio with 𝑟 the side opposite the included angle and 𝑂𝑃 as a hypotenuse. Remember we’ve cut the angle of 60 degrees in half. 60 degrees was equivalent to 𝜋 over three radians. So, half of this is 𝜋 over six radians. And we have sin of 𝜋 over six is equal to 𝑟 over 𝑂𝑃. If we multiply both sides of the equation by 𝑂𝑃, and then divide by sin of 𝜋 over six, we get 𝑂𝑃 is equal to 𝑟 over sin of 𝜋 over six. And since sin of 𝜋 over six is one-half, that tells us that 𝑂𝑃 is equal to two 𝑟.

Substituting two 𝑟 into our first equation, we get 𝑟 is equal to eight minus two 𝑟. We can add two 𝑟 to both sides. And we can divide through by three. And that tells us that the radius of the circular patio is eight-thirds of a yard.

Calculate the total area of grass in one sector. Give your answer in terms of 𝜋 in its simplest form.

We already worked out the total sector area. To find the area covered in grass, we need to subtract the area of the patio from this value. The area of the circular patio is given by the formula 𝜋 multiplied by 𝑟 squared. Since the radius of the circular patio is eight-thirds, this becomes 𝜋 multiplied by eight-thirds squared, which is 64𝜋 over nine. 32𝜋 over three minus 64𝜋 over nine is 32𝜋 over nine. And we’ve worked out the total area of grass that’s in one sector is 32𝜋 over nine square yards.