Video Transcript
In this video, we will learn how to
find the area of a circular segment. We will begin by looking at the
definition of a circular segment. We will then identify formulas that
can be used to calculate the area of a circular segment and then use these to solve
some problems.
What is the definition of a
circular segment? Is it (A) a region of a circle
bounded by an arc and a chord passing through the end points of the arc? (B) A region of a circle
bounded by two radii and an arc. (C) A region of a circle
bounded by a chord and a central angle subtended by that arc. (D) A region of a circle
bounded between two chords and two arcs. Or (E) an arc which is half of
the circumference.
The circle shown has been split
into two segments. The smaller part of the circle
is known as the minor segment. The larger part is known as the
major segment. Both of these segments are
bounded by an arc. They are also bounded by a
common chord. This means that the correct
definition is option (A). A circular segment is a region
of a circle bounded by an arc and a chord passing through the end points of the
arc. In our diagram, we have a minor
segment in orange and a major segment in pink.
We will now look at the formulas
that can be used to calculate the area of a segment.
Which formula can be used to
find the area of a circular segment, given radius 𝑟 and a central angle 𝜃?
We recall that any circle can
be split into a minor and major segment as shown. We’re also told that the radius
of the circle is 𝑟 and the central angle is 𝜃. If we label the two points at
the end of the chord 𝐴𝐵 and the center of the circle 𝑂, then the area of the
segment will be equal to the area of the sector 𝐴𝑂𝐵 minus the area of
triangle 𝐴𝑂𝐵. It is important to note at this
point that our angle 𝜃 might be given in degrees or in radians. 180 degrees is equal to 𝜋
radians. We know that a circle has a
total of 360 degrees, which means it will have a total of two 𝜋 radians. As a result, the area of a
circular segment can be calculated using two linked formulas, one for degrees
and one when the angle is in radians.
When our angle was measured in
degrees, the area of a sector is equal to 𝜃 out of 360 multiplied by 𝜋𝑟
squared. As already mentioned, 360
degrees is equal to two 𝜋 radians. This means that the area of a
sector when 𝜃 is in radians is 𝜃 over two 𝜋 multiplied by 𝜋𝑟 squared. In this case, the 𝜋s
cancel. We are left with 𝜃 over two
multiplied by 𝑟 squared, which is often written as a half 𝑟 squared 𝜃.
As we’ve worked out a formula
for the area of a sector in degrees and radians, we will now look at the area of
a triangle. The area of any triangle can be
calculated using the formula a half of 𝑎𝑏 multiplied by sin 𝐶. In our diagram, we can see that
the lengths 𝑎 and 𝑏 are both equal to the radius or 𝑟. The angle 𝐶 is equal to
𝜃. Therefore, the area of a
triangle inside a circle can be calculated using the formula half 𝑟 squared
multiplied by sin 𝜃. We will now clear some space to
work out the formula that can be used to find the area of a circular
segment.
Let’s consider when 𝜃 is
measured in radians first. The area of the sector is a
half 𝑟 squared 𝜃, and the area of the triangle is a half 𝑟 squared sin
𝜃. We can factor out a half 𝑟
squared as this is common in both terms. Inside the parentheses or
bracket, we’re left with 𝜃 minus sin 𝜃. When the central angle 𝜃 is
given in radians, then the area of the circular segment can be calculated by
multiplying a half 𝑟 squared by 𝜃 minus sin 𝜃. If the central angle is given
in degrees, then our formula is equal to 𝜃 over 360 multiplied by 𝜋𝑟 squared
minus a half 𝑟 squared sin 𝜃.
Whilst the 𝑟 squared is common
in both terms, we tend not to factor it out here but instead calculate the area
of the sector and area of triangle separately. We then subtract our two
answers to calculate the area of the circular segment. Either one of these formulas
can be used depending on the context of the question.
We will now use these to find the
area of a segment given different properties of a circle.
The area of a circle is 227 square
centimeters and the central angle of a segment is 120 degrees. Find the area of the segment,
giving the answer to two decimal places.
We’re told in the question that the
central angle of a segment is 120 degrees. And we need to calculate the area
of this segment. When the angle of a segment is
given in degrees, we can calculate the angle of this segment by subtracting the area
of the triangle from the area of the sector. The area of the sector is equal to
𝜃 over 360 multiplied by 𝜋𝑟 squared. The area of the triangle is equal
to a half 𝑟 squared multiplied by sin 𝜃. We are told in the question that
the area of the circle is equal to 227 square centimeters. This means that 𝜋𝑟 squared equals
227. Dividing both sides of this
equation by 𝜋 gives us 𝑟 squared is equal to 227 over 𝜋.
We can now substitute these into
both of our formulas. The area of the sector is equal to
120 over 360 multiplied by 227. This can be simplified to one-third
multiplied by 227 or 227 over three. The area of the triangle can be
calculated by multiplying a half by 227 over 𝜋 by sin of 120 degrees. Sin of 120 degrees is equal to root
three over two. The area of the segment can
therefore be calculated by subtracting a half multiplied by 227 over 𝜋 multiplied
by root three over two from 227 over three. Typing this into the calculator
gives us 44.37875 and so on. As we need to round our answer to
two decimal places, the key or deciding number is the eight. This means that we round up to
44.38. The area of the segment is 44.38
square centimeters.
As this is the area of the minor
segment, we could calculate the area of the major segment by subtracting this answer
from 227 square centimeters.
We will now look at another
question where we are given the radius and the chord.
A circle has a radius of 10
centimeters. A chord of length 14 centimeters is
drawn. Find the area of the major segment,
giving the answer to the nearest square centimeter.
We are told that the circle has a
radius of 10 centimeters. A chord of length 14 centimeters is
drawn on the circle. If we let the two ends of the chord
be points 𝐴 and 𝐵 and the center point 𝑂, then the area of the minor segment is
equal to the area of the sector minus the area of the triangle. In order to calculate both of
these, we firstly need to work out the central angle 𝜃. This can be done in either radians
or degrees. In this question, we will use
radians. So, it is important that our
calculator is in the correct mode. The area of a sector, when 𝜃 is in
radians, is equal to a half 𝑟 squared 𝜃. And the area of a triangle is equal
to a half 𝑟 squared sin 𝜃. This can be simplified by
factoring, giving us the area of the segment equal to a half 𝑟 squared multiplied
by 𝜃 minus sin 𝜃.
We can now calculate the angle 𝜃
by using right-angle trigonometry or the cosine rule. In order to calculate the angle in
any triangle using the cosine rule, we use the following formula. Cos of 𝐴 is equal to 𝑏 squared
plus 𝑐 squared minus 𝑎 squared divided by two 𝑏𝑐, where 𝑎, 𝑏, and 𝑐 are the
three lengths of the triangle and 𝐴 is the one opposite the angle we’re trying to
work out. Substituting in our values gives us
cos of 𝜃 equals 10 squared plus 10 squared minus 14 squared over two multiplied by
10 multiplied by 10. This simplifies to cos of 𝜃 equals
one fiftieth. Ensuring that our calculator is in
radian mode, 𝜃 is equal to the inverse cos of one fiftieth. This is equal to 1.55079 and so on
radians.
We can now substitute this value
into our formula for the area of a segment. The area of the minor segment is
equal to 27.5497 and so on. We have been asked to calculate the
area of the major segment. This is the area of the whole
circle minus the area of the minor segment. The area of a circle is equal to
𝜋𝑟 squared. As our radius is equal to 10
centimeters, the area is equal to 100𝜋. We need to subtract 27.5497 and so
on from this. This is equal to 286.6095 and so
on. We’re asked to round our answer to
the nearest square centimeter. The deciding number is the six in
the tenths column. So, we round up to 287 square
centimeters. This is the area of the major
segment in the circle.
We will now summarize the key
points from this video. A segment is a region bounded by an
arc and a chord passing through the end points of the arc. Drawing a chord on any circle
splits it into two segments, a minor segment and a major segment. As suggested by the name, the major
segment is the larger of the two parts. The area of the minor segment,
shown in orange on the diagram, is equal to the area of the pink sector minus the
area of the blue triangle. In order to calculate these areas,
we need the angle 𝜃 in either radians or degrees and the length of the radius
𝑟.
When dealing in degrees, we tend to
calculate the two areas separately. The area of the sector is equal to
𝜃 over 360 multiplied by 𝜋𝑟 squared. And the area of the triangle is
equal to a half 𝑟 squared multiplied by sin 𝜃. When dealing with radians, we can
simplify by factoring. The area of the segment when 𝜃 is
in radians is a half 𝑟 squared multiplied by 𝜃 minus sin 𝜃. In any question, we can convert
from one angle measurement to the other using the fact that 180 degrees is equal to
𝜋 radians and, therefore, 360 degrees is equal to two 𝜋 radians. In some questions, we might need to
use trigonometry to find out the angle 𝜃 first.
