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
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
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.