### 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 ππ΄ and ππ΅ are both
tangents to the circle, and the arc π΄π΅ touches the circle at a single point. Work out the area of sector
ππ΄π΅. Give your answer in terms of
π.

Recall, the formula for the area of
a sector with angle π radians and radius π is a half multiplied by π squared
multiplied by π. We canβt currently use this formula
to work out the area of sector ππ΄π΅ since the included angle is in degrees rather
than radians. First, weβll 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 letβs substitute what we know
about the sector into the formula for its area. Itβs a half multiplied by the
radius squared. Thatβs 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.

Thereβre actually two triangles
weβre 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 circleβs radius is π over two radians. We can therefore add in the radius
of the patio forming a right-angled triangle at point π, which is the centre of the
circle. We also know that if we extend the
line ππ up to the circumference of the circle at πΆ, that whole line is eight
yards, since itβs the line joining the centre of the lawn to a point on its
circumference. Itβs the radius of the lawn
itself.

The line segment ππΆ is also the
radius of the circle. Since ππ and ππΆ give a total
length of eight yards, and ππΆ is the radius of the circle, we can say that ππ
plus π is equal to eight. We can subtract ππ from both
sides of this equation, and that tells us that the radius is eight minus ππ. Now letβs 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 ππ.

We can use the sine ratio with π
the side opposite the included angle and ππ as a hypotenuse. Remember weβve 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 π over ππ. If we multiply both sides of the
equation by ππ, and then divide by sin of π over six, we get ππ is equal to π
over sin of π over six. And since sin of π over six is
one-half, that tells us that ππ is equal to two π.

Substituting two π into our first
equation, we get π is equal to eight minus two π. We can add two π 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 π 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 weβve worked out the total area
of grass thatβs in one sector is 32π over nine square yards.