The radius of a circle is 10
centimeters and the perimeter of a sector is 25 centimeters. Find the area of the sector.
Let’s begin with a diagram to
visualize the situation. We have a circle with a radius of
10 centimeters. We’re then told that there is a
sector of this circle with a perimeter of 25 centimeters. Now a sector of a circle is an area
enclosed between two radii. So, each of these lines, which
enclose the sector, are 10 centimeters. The perimeter of this sector is the
distance all the way around its edge. So, that’s the sum of two radii and
then the arc length.
We can use the information we’ve
been given to form an equation. The perimeter is 25 centimeters and
the radius is 10 centimeters. So, we have the equation 25 is
equal to two multiplied by 10 plus the arc length. Two multiplied by 10 is 20. And then subtracting this value
from each side of the equation, we see that the arc length is five centimeters.
So, we know the arc length, but how
does this help us with working out the area of this sector? Well, we know that the area of any
circle can be found using the formula 𝜋𝑟 squared, where again 𝑟 represents the
radius. To find the area of a sector, we
multiply this value by the fraction of the full circle that the sector
represents. So, if the angle at the center of
the sector is 𝜃, then the fraction of the full circle will be 𝜃 over 360, as there
are 360 degrees in a full turn. We know the radius of the circle;
it’s 10 centimeters. So, if we can work out the angle 𝜃
at the center of this sector, then we’ll be able to use this formula to find its
We have a similar formula for
working out an arc length. Two 𝜋𝑟 gives us the circumference
of a full circle. And then we multiply this by the
fraction of the circle that we have, which again is 𝜃 over 360. So, as we know the radius of our
circle, and we’ve worked out the arc length of this sector, we can use this formula
for arc length to work out the angle 𝜃 at the center of the sector, which we’ll
then be able to substitute into our formula for its area. Substituting 10 for the radius and
five for the arc length into this last formula gives 𝜃 over 360 multiplied by two
multiplied by 𝜋 multiplied by 10 is equal to five.
We can solve this equation to find
the value of 𝜃. First, we multiply by 360 to bring
this out of the denominator on the left-hand side, which will give five multiplied
by 360 in the numerator on the right. We also need to divide by two and
𝜋 and 10, which means we’re dividing by 20𝜋. So, we have that 𝜃 is equal to
five multiplied by 360 over 20𝜋. Simplifying the numbers in this
value gives 90 over 𝜋 because 90 is five multiplied by 360 over 20. And if we want to work this out as
a decimal, it’s equal to 28.647 continuing.
So, now that we know the value of
𝜃, the central angle for this sector, as well as the circle’s radius, we can apply
our formula. Our formula, remember, is 𝜃 over
360 multiplied by 𝜋𝑟 squared. Now, we could substitute our
decimal value of 𝜃. But if we use the exact value of 𝜃
as 90 over 𝜋, then we’ll get an exact value as our answer. We can recall that 𝜃 over 360 is
equal to 𝜃 multiplied by one over 360. So, substituting 90 over 𝜋 for 𝜃
and 10 for 𝑟, we have that the area of the sector is equal to 90 over 𝜋 multiplied
by one over 360 multiplied by 𝜋 multiplied by 10 squared.
The factor of 𝜋 in the denominator
of our value for 𝜃 will cancel with the 𝜋 in the numerator of the formula, which
is why we used this exact value. 90 and 360 can also be canceled
because they share a common factor of 90. 90 divided by 90 is one, and 360
divided by 90 is four.
So, now we’re left with one
multiplied by one multiplied by 10 squared in the numerator and just four in the
denominator. 10 squared is 100. So, we have 100 over four. And 100 divided by four is 25. The units for this area will be
centimeters squared. So, we found that the exact area of
this sector is 25 centimeters squared.