Determine the perimeter and the area
of the given figure.
The perimeter is the distance around the figure
and the area is the number of square units
that will fit inside the figure.
Notice how the figure consists of a square
as well as two half circles.
Before we determine the perimeter,
let's review the perimeter formulas shown below.
The perimeter of a circle
is equal to the circumference of a circle,
which is equal to pi times d,
where d is the diameter,
or two times pi times r, where r is the raduis.
Remember the radius is equal to half of the diameter.
And for a square, to find the perimeter,
we sum the lengths of the four sides,
which is equal to four s.
So looking at the figure,
notice how the distance around the outside
is equal to the sum
of the lengths of two sides of the square,
these lengths here,
and then plus the length around this half circle,
which is half of the circumference of a circle,
plus the length around this half of the circle,
or half of the circumference.
So notice how the sum of the lengths around the half circles
is equal to the length around one full circle
or one full circumference,
given by the formula pi times d.
So we can say the perimeter
is equal to the sum of the lengths of two sides,
or two s, from the square,
plus two half circumferences,
which is equal to one full circumference,
which is equal to pi times d.
So s is eight inches, the length of one side of the square
and so is the diameter,
and therefore the perimeter is equal to
two times eight inches
plus pi times eight inches.
And therefore the exact perimeter
is equal to 16 plus eight pi inches.
But we are told to round to two decimal places,
so now we'll go to the calculator.
Notice how we have a two in the third decimal place,
and therefore we round down to 41.13 inches.
And now let's work on determining the area of the figure,
which is the number of square units that will fit
inside the figure.
Before we do this,
the area of a circle is equal to pi times r squared
where r is the radius
and the area of a square is equal to s times s
or s squared,
where s is the side length.
So notice how the area of the figure
is equal to the area of the square,
this area here,
plus the area of the two half circles,
which is this area here.
So the area is equal to the area of the square,
which is equal to s times s, or s squared,
and then plus the area of the two half circles.
Well the area of two half circles
will be equal to the area of one full circle,
given by the formula pi r squared.
So the area of the figure
is equal to s squared plus pi r squared.
Notice to find the area of the two half circles,
or one full circle,
we do need to determine the radius,
which is half of the diameter.
So this length here, of eight inches, is the diameter.
And therefore the radius,
the distance from the center of the circle to the edge,
would be half of the diameter, or four inches.
So we have the area is equal to s squared,
which is the square of eight inches,
plus pi times the square of the radius,
where the radius is four inches.
So we have pi times four squared.
Simplifying, we have the area is equal to,
eight squared is 64,
plus four squared is 16,
pi times 16 is 16 pi.
So the exact area is 64 plus 16 pi square inches.
And now let's find the area rounded to two decimal places.
Going back to the calculator,
we have 64 plus 16 pi.
Notice how we have a five in the third decimal place,
and therefore we round up to 114.27.
So the area is approximately 114.27 square inches.
I hope you found this helpful.