Text to Supplement Lesson on Basic
Area, Perimeter and Volume
In this lesson, we will review area, perimeter and volume. In
particular we will first look at the definitions of basic
2dimensional
shapes: rectangles,
triangles and circles and how to compute the
area and perimeter of that shape. Next we will work out a
problem similar to the type you may expect to see
on the CAHSEE. Finally we will review how to find the
volume of a rectangular solid.
1. Finding the Area and
Perimeter of a Rectangle
Definition: A polygon is a
closed shape bounded only by straight lines. A rectangle
is a 4sided polygon in with all four of its angles are right angles.
Example Rectangle:


To calculate the area of a rectangle multiply its
length and width together. In the above example multiply
3 inches by 6
inches
Area = (3 inches)(6 inches)
= 18 inches^{2}
When computing area our answer is always given in
square units. This is because pictorially what we are doing when we
calculate area is equivalent to dividing up the shape into equal unit
squares and adding them up.
Now try one by yourself. If you want to see the answer, put your mouse
on the yellow rectangle and the answer will appear.
Exercise 1
Find the area of the rectangle shown below
Answer:
The perimeter of a polygon is equal to the
distance all the way around the figure. So to calculate the perimeter of
a square add twice the length and twice the width together.
The perimeter of the above rectangle is
Perimeter = 12 inches + 6 inches
= 18
inches.
On the CAHSEE you might be asked a question about a square. Remember
that a square is a rectangle in which the length and width are the same
length. In this case you only need to know one side length in order to
calculate both the area and the perimeter of the square.
Now try one by yourself. If you want to see the answer, put your mouse
on the yellow rectangle and the answer will appear.
Exercise 2
Find the perimeter of the rectangle shown below
Answer:
2. Finding
the Area and Perimeter of a Triangle
Definition: A polygon is a
closed shape bounded only by straight lines. A triangle
is a 3sided polygon.
Example Triangle:


To calculate the area of a triangle
multiply its base times its height and divide by two. The
triangle above has area
Area = 18 cm x 12 cm x (1/2)
= 108 cm^{2}
Now try one by yourself. If you want to see the answer, put your mouse
on the yellow rectangle and the answer will appear.
Exercise 3
Find the area of the area of the triangle shown below
Answer:
The perimeter of a polygon is equal to the
distance all the way around the figure. So to calculate the perimeter of
a triangle add the lengths if its sides together.
The perimeter of the above triangle is
Perimeter = 18 cm + 15 cm + 15 cm
= 48 cm
Now try one by yourself. If you want to see the answer, put your mouse
on the yellow rectangle and the answer will appear.
Exercise 4
Find the perimeter of the triangle from
Exercise 3 (shown below)
Answer:
You may see a triangle area problem on the CAHSEE in which you do not
know the height of the triangle.
These types of problems require that you know how to use the
Pythagorean Theorem. This will be taught in a
later lesson, but follow the link if you wish to look at that lesson
now.
3. Finding the Area and Circumference of a
Circle
Definition: A
circle is
the set of all points in a plane that are a fixed distance from a given
point in the plane.
This fixed distance is called the radius of the circle. The example
circle below has radius = 25 mm
Example Circle:


This circle has a diameter of 50 mm. The
diameter is the
distance from any point on the circle, through its center, to a point on
the opposite side of the circle. The radius of a circle is equal to half
the diameter.
Radius = 1/2 Diameter
The distance around the outside of a circle has a special name, it is
called the Circumference and is denoted
C.
Pi: There
is a special number called pi that is equal to the ratio of
circumference to diameter on any circle and is approximately
equal to 3.14. We use this number when the calculating area and
perimeter of a circle. The symbol that we use for
pi is
p. The
illustration on the right uses this symbol.


To calculate the area of a circle
multiply 3.14 times the radius,
r, of the circle squared.
Area of a Circle ≈ 3.14
r^{2}
which is the same thing as multiplying 3.14 by the radius and then
again by the radius.
The circle above has area
Area = 3.14 x 25 mm x 25 mm
= 1962.5 mm^{2}
To calculate the circumference of the circle multiple
3.14 by its
diameter. Notice this is the same thing as multiplying
3.14 by two
times the radius of the circle.
The circumference of the above circle is
Circumference = 3.14 x 50 mm
= 157 mm
Now try one by yourself. If you want to see the answer, put your mouse
on the yellow rectangle and the answer will appear.
Exercise 5
Find the area and circumference of the circle shown below.
Answer:
On the CAHSEE you might be given a question involving a circle in
which you are given the diameter but not the radius. To calculate the
radius just divide the diameter in half. For instance, you know that the
diameter in the circle above is 50
mm, so the radius is half of that
number: 25 mm.
You may even get a tricky problem like the one below in which you are
given the circumference but not the diameter or the radius. In this case
you can compute the diameter (and therefore radius) by solving for
d
(the diameter) in the circumference formula.
Tricky Example: Find the area of the circle that
has circumference 25 cm.
Notice in this example that we haven't been given the radius or
the diameter of the circle. What we have been given is that the
circumference is 25cm. From the formula for circumference of a
circle we can solve for d.
C = 3.14 x d
d = C / 3.14
d = 25 cm / 3.14
d = 7.96 cm
The radius is half the diameter, so
r = 7.96/2 = 3.98 cm
or about 4 cm. Now, we can find the area by using
A =
p r^{2}
≈ 3.14 (4^{2})
= (3.14)(16)
= 50.24 cm^{2} 

Now try one by yourself. If you want to see the answer, put your mouse
on the yellow rectangle and the answer will appear.
Exercise 6
Find the area of the circle that has a circumference of
50 inches.
Answer:
4. Typical CAHSEE Type Question
A circular flowerbed with radius 6 ft is inscribed within a grassy
square region.
a. Find the area of the entire
square region. Area of square
= length (l) x length (l)
In this problem we are not given the length of the square but
notice that the length of the radius of the flower bed is the
same as one half the side length of the square. Multiplying the
radius by two will give us length. So, area of square
Area
= 12 ft x 12 ft
= 144 ft^{2} 

b. Now find the perimeter of the grassy region and the flower bed so
that we know how much material needs to be purchased to fence both.
Perimeter of grassy region
= 4 x length (all the way around the square)
= 4 x 12 ft
= 48 ft
Circumference of circular flowerbed
Circumference
= 3.14 x d
= 3.14 x 12 ft
= 37.7 ft
Adding these two together gives us
Total =
Perimeter of Grassy Region + Circumference of Circular Flowerbed
= 85.7 ft.
5. Finding the volume of a Rectangular Solid
Definition: A rectangular solid
is a 3 dimensional object in which each of its
6 faces is a rectangle
and adjacent faces are perpendicular. A rectangular solid has three
dimensions, length, width and height.
Example:


The rectangular solid above has length 3 inches, width
5 inches and
height 2 inches.
Definition: Volume is the
amount of space occupied by a threedimensional object, expressed in
cubic units. To calculate the volume of a rectangular solid you multiple
the area of its base times its height.
So for the example above the volume is length times width (area of
rectangle) times height. The volume of this rectangular solid is
Volume = (3)(5)(2) = 30
inches^{3}
Now try one by yourself. If you want to see the answer, put your mouse
on the yellow rectangle and the answer will appear.
Exercise 7
Find the volume of the rectangular solid shown below.
Answer:
