In this video, we will learn how to
find the area of a circular sector and solve problems that relate this area to the
arc length and perimeter of the sector. We recall first that an arc is a
portion of the circumference of a circle. A sector is a portion of the circle
itself formed by two radii. That’s the lines that connect the
center of the circle to its circumference and an arc. It’s sort of like the surface of a
wedge of cheese cut from a wheel or a slice of pizza. The angle at the center of this
wedge, the angle between the two radii, which we often denote using the Greek letter
𝜃, is called the central angle of the sector.
If this central angle 𝜃 is less
than 180 degrees, we call the sector a minor sector. That’s because there will also be
another sector in the circle whose central angle will be greater than 180
degrees. This sector’s bigger, so we call it
the major sector. Now finding a formula for the area
of a sector is fairly straightforward, but it does depend on whether we’re working
in degrees or radians for the central angle.
Let’s consider a formula in degrees
first of all. We know that the area of a full
circle is calculated using the formula 𝜋𝑟 squared. But when we’re working with a
sector, we only have a portion of the full circle, and so we only want a fraction of
its area. The fraction of the circle we have
is determined by the central angle. It will be 𝜃 over 360 because
there are 360 degrees in a full turn. So to find the area of a sector, we
simply multiply the area of the full circle by the fraction that the sector
represents. So we have 𝜃 over 360 multiplied
by 𝜋𝑟 squared.
So that’s great if we’re working in
degrees. But if we’re working in radians, we
need something different. In our first problem, we’ll
consider how we can derive an alternative formula to use if instead the central
angle is measured in radians.
Write an expression for the area of
a sector whose arc’s measure is 𝜃 radians, knowing that the expression for the area
of a sector measuring 𝜃 degrees is 𝜋𝑟 squared 𝜃 over 360.
So we’re reminded of the formula we
can use to calculate the area of a sector when the central angle is given in
degrees. And we’re asked to use this to
determine a different formula we can use when the angle is given in radians. We should recall that when we’re
working in radians, a full turn, which in degrees is equivalent to 360 degrees, is
two 𝜋 radians. So we can take the formula that we
know for the area of a sector in degrees, and we can replace the 360 in the
denominator, which represents the 360 degrees in a full turn with two 𝜋. Doing so gives 𝜋𝑟 squared 𝜃 over
two 𝜋. Now, of course, we can cancel a
factor of 𝜋 from the numerator and denominator of this fraction, which leaves us
with 𝑟 squared 𝜃 over two or, equivalently, one-half 𝑟 squared 𝜃.
So we’ve used the area of a sector
in degrees to find an expression for the area of sector when the central angle is
given in radians; it’s one-half 𝑟 square 𝜃.
So we now have the two formulae for
determining the area of a sector for central angle measured in either degrees or
radians. When answering a problem, we must
make sure we choose the appropriate formula for the measure that has been used for
the central angle. It will also be helpful at this
point to recall the formulae for calculating an arc length both in degrees and
In degrees first of all, the arc
length is 𝜃 over 360 multiplied by 𝜋𝑑 or two 𝜋𝑟. It’s the fraction of the circle we
have multiplied by the circle’s circumference. And in radians, the arc length is
simply 𝑟𝜃. That’s because if we take the
formula in degrees and then replace 360 with two 𝜋 and 𝑑, the diameter, with two
𝑟, we have 𝜃 over two 𝜋 multiplied by 𝜋 multiplied by two 𝑟. And of course, the 𝜋’s will cancel
and so will the twos, leaving us with simply 𝑟𝜃.
We’ve got all the formulae we
need. So now let’s consider some
examples. The problems that we consider in
this video will be applications of these results to questions which require an
element of problem solving.
The radius of a circle is five
centimeters and the area of a sector is 15 centimeters squared. Find the central angle, in radians,
rounded to one decimal place.
In this problem, we have a sector
of a circle. We know that the radius of the
circle is five centimeters and we know that the area of the sector is 15 square
centimetres. But we don’t know the central
angle. We were asked to find the central
angle in radians. So we need to recall the key
formulae we need. When working in radians, the area
of a sector is given by a half 𝑟 squared 𝜃, where 𝑟 represents the radius of the
circle and 𝜃 represents the central angle. We can therefore use the
information we’re given to form an equation. The area of the sector is 15 square
centimeters, and the radius of the circle is five centimeters. So we have the equation 15 equals a
half multiplied by five squared multiplied by 𝜃. We can now solve this equation to
determine the value of 𝜃.
Firstly, we evaluate five squared,
which is 25. We can then multiply both sides of
the equation by two and then divide both sides by 25, giving 𝜃 equals two
multiplied by 15 over 25. Two multiplied by 15 is 30, and
then we can cancel a factor of five from both the numerator and denominator. So this simplifies to six over
five. Six over five is equivalent to one
and one-fifth or 1.2. And so we have our answer to the
problem. The central angle of the sector in
radians is 1.2 radians.
Notice that in this question it was
important to use the area formula in radians. It would have been possible to
instead use the area formula in degrees and then convert the answer from degrees to
radians at the end. But that would require an extra
step. And of course, we may forget to
convert our answer.
Let’s now consider an example in
which we use our knowledge of the area of a sector to find the area of a related
Find the area of the shaded part of
the quadrant in the diagram in terms of 𝜋.
So we’re looking to find the blue
area in the diagram. We’re told that we have a
quadrant. That’s a quarter of a circle, which
means that this is a sector whose central angle is a right angle. Now, looking carefully at the
diagram, we can see that this blue area is made up of a quadrant where the radius of
25 centimeters from which a quadrant of a smaller circle which has a radius of 17
centimeters has been removed. The area we’re looking for then
will be the difference between the areas of these two quadrants. Now the question asks us to give
our answer in terms of 𝜋, we might be asked to do this for two reasons, either
because an exact answer is required or because we’re expected to answer this problem
without a calculator.
We haven’t been told whether to
work in degrees or radians, so let’s choose to work in degrees. We know that the area of a sector
whose central angle is 𝜃 and whose radius is 𝑟 is 𝜃 over 360 multiplied by 𝜋𝑟
squared. In this problem, the central angle
is 90 degrees, so we have 90 over 360 multiplied by 𝜋𝑟 squared, which simplifies
to one-quarter multiplied by 𝜋𝑟 squared or 𝜋𝑟 squared over four. This, of course, makes sense
because we know that a quadrant is quarter of a circle, so the area of a quadrant
will be quarter of the full circle’s area. For the area of the larger quadrant
first of all then, we have one-quarter multiplied by 𝜋 multiplied by 25
squared. And for the smaller quadrant, it’s
one-quarter multiplied by 𝜋 multiplied by 17 squared.
If we want, we can factor by 𝜋
over four giving 𝜋 over four multiplied by 25 squared minus 17 squared. 25 squared is 625 and 17 squared is
289. 625 minus 289 is 336. So we have 336𝜋 over four. And finally, we can cancel a factor
of four from the numerator and denominator to give 84𝜋. The units for the radius were
centimeters, and so the units for the area will be square centimeters. So we found that the area of the
shaded part of the quadrant in terms of 𝜋 is 84𝜋 square centimeters.
Let’s now consider the problem
where we’ll see how we can find the arc length of a sector if we know its area and
we know its central angle.
The area of a circular sector is
1888 square centimetres and its central angle is 1.7 radians. Find the arc length of the sector
giving the answer to the nearest centimeter.
In this problem then, we have a
circular sector. We know its area is 1888 square
centimeters and we know its central angle, which we can call 𝜃, is 1.7 radians. We want to calculate the arc length
of the sector, which we often denote using the letter 𝑠. The key thing to note is that the
central angle of the sector is measured in radians. So the formulae that we’re going to
use for both the area of the sector and the arc length must be the formulae in
radians. Those are a half 𝑟 squared 𝜃 for
the area of the sector and 𝑟𝜃 for the arc length.
In order to calculate the arc
length then, we need to know both the radius and the central angle. Let’s use the information we were
given about the area to form an equation. The area is 1888 square
centimeters. We don’t know the radius, so we’ll
keep the letter 𝑟. And central angle 𝜃 in radians is
1.7. We can now solve this equation to
work out the radius of the sector, which we’ll then be able to substitute into our
formula for the arc length. We can multiply both sides of the
equation by two and then divide by 1.7 giving 𝑟 squared is equal to 1888 multiplied
by two over 1.7, which as a decimal is 2221.176.
To solve for 𝑟, we take the square
root of each side of this equation, taking only the positive value as 𝑟 represents
a length. That gives 47.129, and we’ll keep
this exact value on our calculator display for now. So we’ve now calculated the radius
of the sector, and all that remains is to calculate the arc length. Using the formula arc length equals
𝑟𝜃, we multiply the value we’ve just calculated by the central angle of 1.7, which
gives 80.119. We were asked to give an answer to
the nearest centimeter, so we round down. Using the known area and the known
central angle of this circular sector then, we’ve calculated that the arc length to
the nearest centimeter is 80 centimeters.
In our final example, we’ll see how
we can solve a different geometric problem by applying these techniques.
Three congruent circles with a
radius of 43 centimeters are placed touching each other. Find the area of the part between
the circles giving the answer to the nearest square centimeter.
So we want to find the area of the
part between these circles. That’s this part here. We know that the circles are
congruent, that’s identical, each with the radius of 43 centimeters. If we join the centers of the three
circles, we have a triangle. Each side of this triangle consists
of two radii. And so, in fact, this is an
equilateral triangle with a side length of two times 43; that’s 86 centimeters. The area we’re looking to find is
inside this triangle. The other areas inside this
triangle, that’s these three areas here, are three congruent circular sectors. So our approach to finding the blue
area is going to be to find the area of the triangle and then subtract the area of
these three sectors.
The triangle, first of all. This is a nonright triangle, so we
need to use the formula a half 𝑎𝑏 sin 𝑐, where 𝑎 and 𝑏 represent two sides of
the triangle and 𝑐 represents the angle between them. Each of the side lengths of our
triangle are 86 centimeters. And as it’s an equilateral
triangle, all of the angles are 60 degrees. So we have a half multiplied by 86
multiplied by 86 multiplied by sin of 60 degrees. sin of 60 degrees is root three
over two, so the area of the triangle simplifies to 1849 root three, which we’ll
keep in its exact form for now.
Now, for the area of the sectors,
the area of each sector working in degrees is 𝜃 over 360 multiplied by 𝜋𝑟
squared, where 𝜃 is the central angle. But we have three of these
identical sectors, so we’ll multiply by three. That would be equivalent then to
the formula 𝜃 over 120 multiplied by 𝜋𝑟 squared, as three over 360 is one over
120. So substituting the angle of 60
degrees and the radius of the sector which, remember, is 43 centimeters, we have 60
over 120 multiplied by 𝜋 multiplied by 43 squared. 60 over 120 simplifies to a half
and 43 squared is 1849. So we have 1849𝜋 over two for the
area of the sectors.
We can now go ahead and evaluate
this using our calculators. Doing so gives 298.159. We’re required to give an answer to
the nearest square centimeter, so we’ll be rounding down. We found then that the area of the
part between the circles, which is the difference between the area of a nonright
triangle and three circular sectors, is 298 square centimeters.
Let’s now review some of the key
points that we’ve covered in this video. A circular sector is a portion of a
circle formed by two radii and an arc. We use the Greek letter 𝜃 to
denote the angle between the two radii, which is called the central angle of the
sector. If the central angle is less than
180 degrees or 𝜋 radians, we have a minor sector, whereas if it’s greater than 180
degrees, that’s greater than 𝜋 radians, we have a major sector. The formula we use for calculating
the area of a sector depends on whether the central angle is measured in degrees or
radians. In degrees, it’s 𝜃 over 360
multiplied by 𝜋𝑟 squared, whereas in radians it’s a half 𝑟 squared 𝜃.
We also need to recall the formulae
for calculating an arc length. In degrees, it’s 𝜃 over 360
multiplied by two 𝜋𝑟 or 𝜋𝑑. And in radians, it’s simply
𝑟𝜃. We often use the letter 𝑠 to
denote the arc length. If we wanted to calculate the
perimeter of a circular sector as opposed to just the arc length, we’d need to add
on twice the radius. So we have that the perimeter of
the sector is equal to 𝑠 plus two 𝑟. We can use these formulae to solve
problems relating arc length and the area of a sector as well as problems in other
geometric and real-world contexts.