In this video, we will learn how to
find the arc length and the area of a sector given the angle measure in degrees.
Now, you may be aware of an
alternative way of measuring angles using radians. And in this case, the methods and
formulae needed are a little different. So, it’s important to remember that
throughout this video, we will be working solely with angles given in degrees. Firstly, let’s remind ourselves
what we mean by arcs and sectors. If we have a circle and then we
draw in two radii, the portion of the circle enclosed by these two radii and part of
the circle circumference is called a sector. The portion of the circumference
itself is called an arc. The angle formed between the two
radii, which create the sector, is called the central angle, and this will be any
value up to 360 degrees.
We also recall that the measure of
an arc is defined to be equal to the central angle of its sector. Unless we divide the circle exactly
in half, we will always have a minor sector and a minor arc with a central angle
less than 180 degrees and a major sector and major arc with a central angle greater
than 180 degrees. Let’s think then about how we can
calculate the length of an arc with a central angle of 𝜃 degrees. We know that to calculate the
circumference of a circle, we use the formula 𝜋𝑑 or sometimes two 𝜋𝑟.
When we have an arc, though, we
need to consider that this is only part of that circumference of a full circle, and
so its length will be some fraction of the total. The fraction we have will be
determined by the measure of the central angle. We know that there are 360 degrees
in a full turn. So whatever the measure of the
central angle is, which we can think of as 𝜃, we will have this angle out of 360 as
the fraction of the full circle. We can, therefore, use the formula
𝜃 over 360 multiplied by 𝜋𝑑 to calculate the length of an arc with a central
angle of 𝜃 degrees.
In the same way, we can find the
area of a given sector. This will be part of the area of
the full circle, which is given by 𝜋𝑟 squared. Again, the fraction of the circle
we have will be 𝜃 out of 360. So to find the area of a sector, we
can multiply the area of the full circle by this fraction, given the formula 𝜃 over
360 multiplied by 𝜋𝑟 squared. Again, we should remember at this
point that these formulae are only valid if the angle 𝜃 is given in degrees because
we’ve used the fact that there are 360 degrees in a full turn.
Now that we’ve identified the
formulae we’ll need, let’s look at some examples.
Determine the length of arc 𝐴𝐵𝐶
to the nearest hundredth, given that the circle has radius nine.
We are asked then to calculate the
length of this arc which we recall is part of the circumference of the full
circle. The formula for calculating our arc
length is 𝜃 over 360 multiplied by 𝜋𝑑, where 𝜃 is the central angle of the
sector. What we’re doing here is
calculating the circumference of the full circle and then multiplying by the
fraction of the circle that we have.
We’re told that the radius of this
circle is nine units, and this in turn means that the diameter of the circle, which
is always twice the radius, will be 18 units. The central angle of this sector is
given as 72 degrees. So, substituting 72 for 𝜃 and 18
for 𝑑, we have that the length of arc 𝐴𝐵𝐶 is 72 over 360 multiplied by 18𝜋. Now, we can actually simplify this
because 72 is a factor of 360 and the fraction simplifies to one-fifth. So if we were looking to give an
exact answer to this problem, we could give our answer as 18 over five multiplied by
𝜋 or eighteen-fifths of 𝜋.
However, the question asks us to
give our answer to the nearest hundredth, so we’ll use our calculator to evaluate
this. We get 11.30973. And then rounding to the nearest
hundredth, we have 11.31. There were no units given for the
radius in the questions, so there are no units for our answer. But there would be some kind of
length units, such as centimeters or feet. Our answer to the problem is
In our next example, we’ll see how
we can work backwards from knowing the length of an arc to determining the arc’s
measure or the central angle of the sector.
An arc on a circle with a radius of
50 has a length of 115. Determine the arc’s measure to the
nearest tenth of a degree.
Let’s summarize what we know on a
quick sketch of this problem. We have circle with a radius of 50
units. An arc which is part of the
circumference of this circle has a length of 115 units. We’re then asked to determine the
arc’s measure, which we recall is equal to the central angle of the sector. We can use the Greek letter 𝜃 to
represent this angle. Now, we recall that the formula for
calculating an arc length is 𝜃 over 360 multiplied by 𝜋𝑑. We multiply the circumference of
the full circle by the fraction of the circle that we have.
We can substitute the values we
know into this formula to give an equation. The arc length is 115. We don’t know the value of 𝜃, so
we’ll leave that as it is. And the radius of the circle is 50,
which means that the diameter, which is twice the radius, will be 100. We, therefore, have the equation
115 equals 𝜃 over 360 multiplied by 100𝜋. And we can solve this equation to
determine the value of 𝜃. First, though, we can simplify if
we wish. On the right, the fraction 100 over
360 can be simplified to five over 18 by dividing both the numerator and denominator
To solve for 𝜃, we need to divide
both sides of this equation by five 𝜋 over 18. And we recall that in order to
divide by a fraction, we multiply by the reciprocal of that fraction. So, we have 115 multiplied by 18
over five 𝜋 is equal to 𝜃. Again, we can simplify a little if
we wish by canceling a factor of five from the numerator and denominator to give 𝜃
equals 23 multiplied by 18 over 𝜋.
Evaluating this on a calculator
gives 𝜃 equals 131.78029. And we then need to round this
value to the nearest tenth, which gives 131.8 degrees. This gives the central angle of the
sector. But remember, by definition, it
also gives the measure of the arc. The length of the arc is 115
units. The measure of the arc is 131.8
Let’s now look at another example
in which we’ll combine our knowledge of arc length with some other mathematical
Rectangle 𝐵𝐶𝑀𝐷 is drawn inside
a quarter-circle, where 𝐵𝐷 equals nine centimeters and 𝐵𝐶 equals 12
centimeters. Find the length of arc 𝐴𝐵𝐸,
giving the answer in terms of 𝜋.
So, we’ve been asked to find the
length of the arc 𝐴𝐵𝐸. We know that we have quarter of a
circle, which means that this arc length will be quarter of the circle’s full
circumference. So, it’s equal to one-quarter
multiplied by 𝜋𝑑. We can also think of that value of
one-quarter as 90 over 360 if we wish because this quarter-circle is also a sector
of a circle with a central angle of 90 degrees.
In order to answer this problem
then, we need to know the diameter or perhaps the radius of this circle. Instead, we’ve been given some
other measurements in the question. 𝐵𝐷 is nine centimeters and 𝐵𝐶
is 12 centimeters, but neither of these are the radius or diameter. We can identify three radii of this
circle in the diagram: the lines 𝑀𝐴, 𝑀𝐵, and 𝑀𝐸. Now, as 𝐵𝐶𝑀𝐷 is a rectangle, we
know that the length of the line segment 𝐶𝑀 will be the same as the length of the
line segment 𝐵𝐷. So, it’s also nine centimeters.
If we look carefully at the figure,
we now see that we have a right triangle, triangle 𝐵𝐶𝑀, in which we know the
lengths of two of the sides; they’re 12 centimeters and nine centimeters. And the third side is the radius of
the circle, which we wish to calculate. If we know two sides in a right
triangle and we want to calculate the third, we can do this using the Pythagorean
theorem, which tells us that the sum of the squares of the two shorter sides, 𝑎 and
𝑏, is equal to the square of the hypotenuse. So, this is often written as 𝑎
squared plus 𝑏 squared equals 𝑐 squared.
In our triangle, the hypotenuse is
the radius of the circle because it’s the side directly opposite the right
angle. So, we have 𝑎 and 𝑏 as nine and
12 centimetres and 𝑐 as 𝑟 centimeters. We can, therefore, form an
equation, 𝑟 squared equals 12 squared plus nine squared. And we can solve this equation to
find the radius of the circle. 12 squared plus nine squared,
that’s 144 plus 81, is 225. 𝑟 is, therefore, equal to the
square root of 225, which is 15.
Now, you may actually have already
realized this because nine, 12, 15 is an example of a Pythagorean triple. In fact, it’s an enlargement of the
three, four, five Pythagorean triple with a scale factor of three. And the three, four, five triple is
probably the most easily recognizable. In any case, we now know the radius
of the circle; it’s 15 centimeters. We can, therefore, calculate the
diameter of the circle by doubling this value. The diameter is 30 centimeters.
So, all that remains then is to
substitute this diameter into our formula for the arc length, which remember was
one-quarter multiplied by 𝜋𝑑. We have one-quarter multiplied by
30𝜋. And we can then simplify by
canceling a factor of two from the numerator and denominator. We’re left with 15 over two 𝜋,
which we can write using decimals, if we wish, as 7.5𝜋.
The question asks us to leave our
answer in terms of 𝜋. So, we can include the units of
centimeters and we have our answer to the problem. The length of arc 𝐴𝐵𝐸 is 7.5𝜋
centimeters. Notice that at no point in this
question did we need a calculator. We left our answer in terms of
𝜋. And in our work with the
Pythagorean theorem, we were working with a Pythagorean triple.
We’ve now seen three examples of
calculating arc length. Let’s consider an example where
we’re asked to calculate the area of a circular sector.
The area of a circle is 160 square
centimeters and a central angle of a sector is 71 degrees. Find the area of the sector giving
the answer to two decimal places.
In this problem, we have a circle
whose area we know and a sector with a central angle of 71 degrees whose area we
wish to calculate. We recall that in order to
calculate the area of a sector with a central angle of 𝜃 degrees, we take the area
of the full circle, 𝜋𝑟 squared, and multiply it by the fraction of the circle that
our sector represents; that’s 𝜃 over 360.
In this problem though, we don’t
need to calculate the area of the circle using 𝜋𝑟 squared because we’ve been given
it. So, we can just substitute the
relevant information. 𝜃 is 71 degrees. So, we have 71 over 360 multiplied
by the area of the circle which is 160. We could evaluate this on our
calculators or we could first simplify by canceling a factor of 40 from the
numerator and denominator to give 71 over nine multiplied by four.
Now, we can evaluate and it gives
284 over nine or 31.5 recurring. Rounding to two decimal places and
our value is 31.56. And the units for this area will be
the same as the units for the area of the original circle; they’re square
centimeters. So by multiplying the area of the
circle by the fraction of the circle that this sector represents, we’ve found that
the area of the sector to two decimal places is 31.56 square centimeters.
Let’s now consider one final
example involving the areas of sectors.
In this figure, the diameter of the
larger circle is 41 centimeters and both circles have the same center. Determine, to the nearest tenth,
the area of the shaded part.
So in this question, we have two
concentric circles. That means they have the same
center. And we’re looking to find the area
of the shaded part, which looks a little bit like part of a ring or donut shape. There are lots of different ways
that we could approach this question. We need to think carefully about
what the most efficient approach is. We have these two lines in pink,
which are dividing each circle into a minor segment and a major segment.
I think the easiest way to approach
this problem is as the difference between the area of the major sector of the large
circle, which is outlined in pink, and the major sector of the smaller circle, which
is outlined in green. We know that the area of a sector
can be found by multiplying the area of the full circle, 𝜋𝑟 squared, by the
fraction 𝜃 over 360, where 𝜃 is the central angle of the sector. Both of our sectors share a common
central angle which we need to calculate.
We’ve been given the measure of the
minor arc of the larger circle; it’s 134 degrees. And we recall that the measure of
an arc is equal to the central angle of its sector. So, the central angle of the minor
sectors is 134 degrees. The central angle of the major
sectors then is 360 degrees minus this value which is 226 degrees.
Next, we need to determine the
radius of each of our circles. We’re given that the diameter of
the larger circle is 41 centimeters. So, its radius will be half of
this; that’s 20.5 centimeters. Looking at the figure, we can work
out the radius of the smaller circle as the difference between 20.5 centimeters and
the measurement we’re given of 12.3 centimeters, which is the difference between the
radii of the two circles. The radius of the smaller circle
is, therefore, 8.2 centimeters. And now, we have all the
information we need in order to calculate this area.
The area of the larger major sector
is 226 over 360 multiplied by 𝜋 multiplied by 20.5 squared. And the area of the smaller major
sector is 226 over 360 multiplied by 𝜋 multiplied by 8.2 squared. If we wish, we can factor this
calculation by 226 over 360 multiplied by 𝜋 before evaluating. Using a calculator, we obtain
696.21410. And rounding to the nearest tenth,
we have our answer to the problem. The area of the shaded part of the
figure, which is the difference in the areas of the major sectors of the two
circles, is 696.2 square centimeters.
Let’s now summarize some of the key
points from this video. Firstly, an arc is a portion of the
circumference of a circle, and a sector is a portion of the circle itself enclosed
by two radii and an arc. The measure of an arc is defined to
be the central angle of the sector, here represented using the Greek letter 𝜃. To calculate an arc length, we
multiply the full circumference of the circle by the fraction of the circle that we
have, giving 𝜃 over 360 multiplied by 𝜋𝑑.
We calculate the area of a sector
in a similar way, this time multiplying the area of the circle 𝜋𝑟 squared by the
fraction of the circle we have, 𝜃 over 360. Finally, we must remember that the
formulae used in this video can only be applied if the central angle 𝜃 is measured