### Video Transcript

In this video, we will learn how to
find the area and the circumference of a circle given its radius or diameter and how
to relate both the area and circumference to solve various problems.

Letβs begin by recapping some of
the key language associated with circles. A circle is a set of points that
have a constant distance from a point in the center, here denoted by the letter
π. A radius of a circle is any line
segment that connects the center of the circle to its edge, here labeled using the
letter π. A diameter of a circle is a line
segment whose endpoints both lie on the edge of the circle and that passes through
the center. We denote the diameter using the
letter π and note that it is always twice as long as the radius of the same
circle.

The circumference of a circle is
the distance all the way around its edge. And we denote this using the letter
πΆ. Circles are very special shapes
because their properties are related by a mathematical constant, which we call π
and denote using the Greek letter. π is approximately equal to 3.142
to three decimal places. But in fact, π is what is known as
an irrational number, which means it has an infinitely long string of digits after
the decimal point with no repeating pattern. If we tried to write all these
digits down, weβd be trying forever.

The first key formula relating to
circles that we need to know is this. The circumference of a circle is
equal to π multiplied by the diameter. πΆ is equal to ππ. As the diameter is twice the
radius, this can also be written as πΆ is equal to two ππ.

Letβs begin by considering an
example in which we calculate the circumference of a circle given the length of its
diameter.

Work out the circumference of the
circle, giving your answer accurate to two decimal places.

We recall first that the
circumference of a circle is the distance all the way around its edge. The formula for calculating the
circumference of a circle is πΆ equals ππ, where πΆ represents the circumference
and π represents the circleβs diameter. Looking at the diagram weβve been
given, we can see that this line segment is a diameter of the circle because its two
endpoints each lie on the circleβs edge and it passes through the center of the
circle. The length of this line is six, so
the value of π is six. The circumference of the circle is
therefore π multiplied by six, which we can write as six π.

Now if we needed to give an exact
answer or if we didnβt have access to a calculator, we could leave our answer in
this form. But weβre asked to give our answer
accurate to two decimal places. So we need to evaluate this. Because π is an irrational number,
we get an answer that is also irrational, 18.84955 continuing. Rounding to two decimal places
gives 18.85.

We werenβt given any units for the
diameter in the question, so we donβt have any units for the circumference. But as the circumference is a
measure of length, the units will be general length units. We found that the circumference of
the circle to two decimal places is 18.85.

So weβve now seen how to calculate
the circumference of a circle by applying this formula. We can also apply these formulae to
find the perimeter of other shapes related to circles. For example, suppose we had a
semicircle. The perimeter of this figure is
composed of two parts: a straight edge, which is the length of the circleβs
diameter, and this curved portion, which we refer to as an arc.

As this is a semicircle, the length
of this arc will be half the circumference of the full circle. So using the formula weβve already
written down, it will be ππ over two. If weβre finding the total
perimeter of this semicircle as opposed to just the curved length, we must remember
to add on the straight edge. So we have that the perimeter of a
semicircle, represented by π, is equal to π plus ππ over two.

We can apply the same reasoning to
find a formula for the perimeter of a quarter circle. This time, we have two straight
edges, each of which are radii of the original circle. We then have a curved edge which is
quarter of the circleβs full circumference, so its length will be ππ over
four. The perimeter of the quarter circle
is therefore equal to two π plus ππ over four. And as two π is equal to π, we
can also write this as π plus ππ over four.

We can also express each of these
formulae in terms of πΆ, the circumference of the original circle. For the semicircle, the perimeter
is equal to π plus πΆ over two. And for the quarter circle, the
perimeter is π plus πΆ over four. Letβs now consider an example
involving portions of circles.

Using 3.14 to approximate π and
the fact that π΄π΅πΆπ· is a square, calculate the perimeter of the shaded part.

At first, it may seem that the
perimeter of this shaded region will be tricky to work out as itβs an unusual
shape. If we look carefully though, we see
that each of these unshaded portions are quarter circles. The shaded region is enclosed by
the curved portions of these four quarter circles. Each of these arc length is
one-quarter of the circumference of a circle, but as there are four of them,
together they form the full circumference of a circle.

We know that the formula for
calculating the circumference of a circle is πΆ equals ππ, where π represents the
circleβs diameter. So the question is, what is the
diameter of this circle?

Considering the quarter circle in
the bottom left of the figure, we can see that the radius of this circle will be
half of the side length of the square. Thatβs 68 over two. So the radius is 34
centimeters. The diameter of a circle is twice
the radius, so the diameter is two times 34; itβs 68 centimeters. In fact, we could have deduced this
from the figure without halving and then doubling again. Two of the radii of the quarter
circles lie along the side length of the square, so twice the radius is 68
centimeters. And as the diameter is twice the
radius, we find again that the diameter is 68 centimeters. Weβve already said that the
perimeter of the shaded region is equal to the circumference of the full circle made
up of these four quarter circles.

So using the formula πΆ equals
ππ, we find that the perimeter of the shaded region is π multiplied by 68. Weβre told in the question to use
3.14 as an approximation for π. So multiplying gives 213.52. The units for this perimeter are
the same as the length units used in the question. So we find that the perimeter of
the shaded part using 3.14 as an approximation for π is 213.52 centimeters.

Letβs now consider the area of
circles. The area is also related to the
circleβs dimensions by the constant π. In this case, the relationship is
that the area is equal to π multiplied by the radius squared. Area equals ππ squared. As the radius is half the diameter,
we could equivalently write this as π multiplied by π over two squared, or ππ
squared over four, but this is much less common. Usually, if weβre given the
diameter of a circle, we would simply divide this value by two to calculate the
circleβs radius and then use the first version of the formula. Letβs now consider an example in
which we apply this formula to find the area of a circle.

Using 3.14 as an approximation for
π, find the area of the circle.

We begin by recalling that the
formula for finding the area of a circle is ππ squared, where π represents the
radius of the circle. From the figure, we identify that
the radius of this circle is 12 centimeters because weβve been given the length of a
line segment connecting the center of the circle to a point on the circleβs
circumference.

So substituting 12 for π, we have
that the area of this circle is equal to π multiplied by 12 squared. We must remember that it is just
the 12 weβre squaring. Weβre not also squaring π. So this is equal to π multiplied
by 144. Weβre told that we should use 3.14
as an approximation for π. So evaluating 3.14 multiplied by
144 gives 452.16. The units for area are square
units.

So we found that the area of this
circle, which has a radius of length 12 centimeters, is approximately 452.16
centimeter squared.

Just like we considered the
perimeters of semi- and quarter circles, we can also find formulae for the areas of
these shapes. And in fact, itβs a little more
straightforward this time because we just need to divide the area of the full circle
by two for a semicircle or by four for a quarter circle. So we have that the area of a
semicircle is ππ squared over two and the area of a quarter circle is ππ squared
over four. Letβs now consider an example in
which we relate the area and the perimeter of a semicircle.

The area of the given semicircle is
51.04 centimeters squared. Find the perimeter of the
semicircle to the nearest centimeter.

We recall that the formula for
finding the area of a semicircle is ππ squared over two, where π is the radius of
the circle. ππ squared gives the area of the
full circle, and then we divide it by two. The perimeter of a semicircle is
made up of a straight edge, which is the circleβs diameter, and a curved portion,
which is half of the circleβs circumference. So the perimeter of the semicircle
is equal to π plus πΆ over two. And the formula for finding the
circumference of a circle is ππ, π times the diameter. The perimeter of a semicircle can
be written as π plus ππ over two.

In order to determine the perimeter
of this semicircle then, we first need to calculate its diameter. Using the given area of the
semicircle and the formula for calculating the area of a semicircle, we can form an
equation: ππ squared over two equals 51.04. If we can solve this equation to
determine the value of π, we can then calculate the diameter of the circle by
recalling that the diameter is twice the radius. We begin by multiplying both sides
of the equation by two to give ππ squared equals 102.08.

Next, we divide both sides of the
equation by π, giving π squared equals 102.08 over π, and then take the square
root of each side of the equation. Evaluating gives π equals 5.7002
continuing. So we found the radius of the
circle, and we can double it to find the diameter, which gives 11.4005
continuing.

Weβre now able to substitute this
value of π into our formula for the perimeter of a semicircle. And we need to make sure we keep
this value as exact as possible. You may want to save this value in
the memory of your calculator so that you can then recall it directly when
evaluating the perimeter. Substituting this exact value of π
and then evaluating gives 29.3084. Weβre asked for the perimeter to
the nearest centimeter, so we round down to 29. By recalling the formulae for
calculating both the area and perimeter of a semicircle then, we found that the
perimeter of this semicircle to the nearest centimeter is 29 centimeters.

Letβs now consider one final
example in which weβll apply our knowledge of circles to a real-world problem.

A goat is tethered by a 10 meter
long rope to the corner of a barn. What area of the field can the goat
reach?

First, we need to think practically
about how the goat can move around this field. If the goat was just tethered
somewhere in the middle of the field, then this would be a much more straightforward
problem. The goat could roam around anywhere
within a circle of radius 10 meters. But because, instead, the goat is
tethered to the corner of the barn, at some point the barn is going to get in the
way.

Suppose first that the goat walks
as far as they can along the top wall of the barn. As the rope is 10 meters long but
this wall is 12 meters long, the goat wonβt be able to get all the way to the
end. So theyβll reach a point 10 meters
away from the other corner before they run out of rope. If the rope stays fully stretched,
the goat can then start walking in a circle centered at the corner of the barn with
a radius of 10 meters.

In fact, we can trace out
three-quarters of the circle until the goat arrives at this point here. But letβs think about what happens
next. At this point, the rope is going to
become caught on the bottom-right corner of the barn, and the goat is going to have
to pivot about this point. Five meters of the rope will be
used up because it would now be lying flat along the eastern edge of the barn. And thatβs the length of this
edge. So the goat only has five meters of
rope left with which to pivot. The goat can therefore walk in a
smaller circle with a radius of five meters until they reach the southern edge of
the barn. Of course, all this is what happens
if the goat stays right at the end of the rope, so this is tracing out the perimeter
of the region the goat can cover.

The goat can also walk everywhere
inside this region if the full length of rope isnβt used. So what we found then is that the
total area the goat can cover is three-quarters of a circle with radius 10 meters
and one-quarter of a circle with radius five meters.

We recall that the formula for
calculating the area of a circle is π multiplied by its radius squared. So for the three-quarter circle,
the area is π multiplied by 10 squared multiplied by three-quarters. And for the quarter circle, the
area is π multiplied by five squared multiplied by one-quarter. Evaluating gives 300 over four π
and 25 over four π. The total area of the field that
the goat can reach then is the sum of these two values, which is 325 over four π
square meters.

Letβs now summarize the key points
from this video. The circumference of a circle is
equal to π multiplied by the diameter, or two π multiplied by the radius. The area of a circle is ππ
squared. The perimeter of a semicircle is
equal to the diameter plus half the circumference, and the area is half the area of
the full circle. Itβs ππ squared over two. The perimeter of a quarter circle
is equal to the circleβs diameter plus one-quarter of the circleβs
circumference. And the area is one-quarter of the
area of the full circle. Itβs ππ squared over four. Given either the radius, diameter,
circumference, or area of a circle, we can calculate any other of these measures
using these formulae.