Perimeters, areas and
other measurements
• In many careers it is
essential to have the ability
to recognize 2-dimensional
images as 3-dimensional
objects. House builders
use 2-dimensional plans to
construct a 3-dimensional
house.
Perimeters and Areas
How many measurements would it take to
find the length of a wall in the classroom?
One
This is one-dimensional measurement. (linear)
How many measurements would it take to
find the size (area) of the floor?
Two
This is two-dimensional measurement. (length
and width)
Perimeter is the distance around a geometric
figure. Perimeter is a linear measurement, in
other words it is one-dimension (length). To
find the perimeter P of a polygon, you can add
the lengths of its sides.
Properties of a rectangle
Opposite sides of a rectangle
are the same length
(congruent).
The angles of a rectangle are
all congruent (90°).
Remember that a 90° angle is
called a “right angle.” So, a
rectangle has four right
angles.
Opposite angles of a rectangle
are congruent.
Opposite sides of a rectangle
are parallel.
You can find the
perimeter of a
rectangle by adding
the lengths of its
sides, Or, since
opposite sides of a
rectangle are equal
length, you can find
the perimeter by using
the formula:
P = 2l + 2w
Area:
the number of square units in a
figure.
People talk about the area of all kinds of
things. Area is always measured in some size
squares. They can be smaller or bigger,
depending on the unit of measurement. Here
are two different types of measurement,
kilometers and inches.
Area is the number of square units needed to
cover a figure.
Here we don’t have any particular unit for
measuring the lengths of the sides of the
rectangle. The sides are 7 and 4. Can you
figure out how many squares is the area?
Dimensions are
measured in squares
so using the proper
unit is important.
yd2 cm2 miles2
Area of a square
A = lw or A = S²
Area of a rectangle
A = lw or A = bh
The area A of a rectangle is the product of its
length (l) and its width (w)
A = lw or A = bh
Find the area of a rectangle:
Some
problems may require you to find an additional
piece of information before finding the area.
This problem expects you to use the
Pythagorean Theorem to find the base of the
rectangle before finding the area.
Area of a parallelogram
A = bh
The base of the parallelogram is the length of
the rectangle. The height of the
parallelogram is the width of the rectangle.
The area A of a parallelogram is the product
of its base b and its height h.
A = bh
The height of a
parallelogram is not the
length of its slanted side.
The height of a figure is
always perpendicular to
the base.
Find the area of a parallelogram:
When working with parallelogram problems, be
sure the height you are using is in fact
perpendicular (makes a right angle) to the
base (side) you are using. In this problem, 8 is
the base and 9 is the height. The side of 10 is
not used in this area.
Area of a triangle
A = ½bh
We have discussed how a square, rectangle or
parallelogram can be divided by drawing a diagonal
from one corner to the opposite corner. This forms
two congruent triangles.
Finding the area of a triangle from here is fairly
simple, just take ½ of the area of the square,
rectangle, or parallelogram, So,
the area A of a triangle is half the product of its
base b and its height h
A = ½ bh
Find the area of the triangle:
It may be necessary, when working with an obtuse
triangle, to look outside the triangle to find the
height. Notice how the height is drawn to an
extension of the base of the triangle.
Remember, the height of a triangle or quadrilateral is a line
perpendicular from the base to the opposite vertex or side.
Some more triangles and their measures. Just
a note to remember, the height of a figure is
always perpendicular to the base.
What do you have to
know before you can
find the area of the
triangle to the left?
Use the technique we
just discussed and
determine the area
of the triangle to the
left.
The length of the
height is 4 and the
length of the base is
(AC) 8.
Now multiply
(4 · 8) ⁄ 2 = 16 units2
Area of a trapezoid
A = ½h(b¹ + b²)
A parallelogram can be divided into two
congruent trapezoids. The area of each
trapezoid is one-half the area of the
parallelogram.
The two parallel sides of a trapezoid are its
bases. If we call the longer side b1 and the
shorter side b2, then the base of the
parallelogram is (b1 + b2).
Find the area of the trapezoid:
When working with a trapezoid, the height
may be measured anywhere between the two
bases. Also, beware of “extra” information.
The 35 and 28 are not needed to compute this
area.
Trapezoid formula:
Remember, perimeter is the distance around a
figure and is measured in linear units.
Area is the space inside a figure and is
measured in square units.
Composite Figures
Several shapes in one
Composite shapes offer a unique challenge.
They can be several basic shapes together
that make up one larger shape.
Find the perimeter and area of this figure. Do
you have enough information? Sometimes you
need to use several different formulas to
complete the problem.
Find the perimeter and area of the
figure.
Composite figures drawn on coordinate grids
are easily label with dimensions.
Draw the figure and find the perimeter and
area.
Find the perimeter
and area of the
figure.
Find the perimeter and
area of the figure.
Pythagorean
Theorem
Using formulas to find
measurements
The Egyptians measured their fields with lengths of
knotted rope. The size of the farmer’s field is used to
work out how big his yield would be, and how much tax
he should pay. This knotted rope indicates a triangle
with sides of length 3, 4, and 5 units.
• Pythagoras’s theory is all
about right-angled triangles.
If squares are constructed
on three sides of a right
angled triangle, then these
squares have a simple but
very important connection.
Right angles and rightangled triangles have
been at the forefront
of building since
ancient times and is
still used today.
Pythagorean Theorem In any right triangle, the sum of the squares
of the lengths of the two legs is equal to the
square of the length of the hypotenuse.
Pythagorean Theorem
a² + b² = c²
What does that mean?
It means that if I
take leg a which is 3 units and square it (3²) and add it
to leg b which is 4 units and square it (4²), that
together they will equal the square of the hypotenuse c
(5²)
a² + b² = c²
3² + 4² = 5²
9 + 16 = 25
25 = 25
Did you know?
Sailboats have triangular
sails to capitalize on wind
at 90° to the boat,
thereby increasing its
maneuverability.
First, let’s identify
the parts of a right
triangle.
The legs are the two
shorter sides of the
triangle, and the
hypotenuse is the
longest side. The
hypotenuse is the side
that is always opposite
the right angle.
Find the length of
the hypotenuse.
Remember, the hypotenuse is the
side opposite the right angle.
The points form a
right triangle with
a = 8 and b = 6
a2 + b2 = c2
82 + 62 = c2
64 + 36 = c2
√100 = √c2
10 = c
(0, 9)
(0, 1)
(6, 1)
Find the length of the
hypotenuse.
a2 + b2 = c2
82 + 152 = c2
64 + 225 = c2
289 = c2
√289 = √c2
17 ≈ c
Find the length of leg b.
a2 + b2 = c2
62 + b2 = 112
36 + b2 = 121
b2 = 121 – 36
b2 = 85
√b2 = √85
b ≈ 9.219544457…
Using Pythagorean
Theorem to find area
a2 + b2 = c2
a2 + 82 = 122
a2 + 64 = 144
a2 = 144 – 64
√a2 = √80
A ≈ 8.94427191
A = ½bh
A = ½(16)(8.9)
A = 71.2 units2
a2 + b2 = c2
a2 + 122 = 132
a2 +144 = 169
a2 = 169 – 144
a2 = 25
√a2 = √25
a=5
A = bh
A = (12)(5)
A = 60 units2
P = 2l + 2w
P = 2(12) + 2(5)
P = 24 + 10
P = 34 units
Find the area and
perimeter of the
rectangle.
Use the Pythagorean Theorem to
find the height of the triangle and
the distance across the pond. Could
you use the height of the triangle to
find the area?
The Pythagorean Theorem can be a
very useful tool to find
measurements.
Find the length between the two points, (1, 5),
(3, 1).
Solving word problems
A jogger is taking his normal
run for exercise. He leaves
home and jogs 8 miles north,
then turns and jogs 5 miles
west. If he decides to jog
straight home, what is the
shortest distance he must
travel to return to his
original starting point.
Pythagorean Triples:
The distinction between the Pythagorean
Theorem and its converse are sometimes over
looked.
The Pythagorean Theorem states that if a
triangle is a right triangle, then the lengths of
the sides satisfy the equation a²+ b²= c².
The converse says that if you have three
numbers that satisfy the equation a²+ b² = c²,
then those three numbers are side lengths of
a right triangle.
The most common special sets of triples are
below.
3, 4, 5
a²+ b² = c²
3² + 4² = 5²
9 + 16 = 25
25 = 25
5, 12, 13
a² + b² = c²
5² + 12² = 13²
25 + 144 = 169
169 = 169
8, 15, 17
a² + b² = c²
8² + 15² = 17²
64 + 225 = 289
289 = 289
Other triples do not work. Remember the
three numbers must make the equation
a² + b² = c² true.
7, 8, 9
a² + b² = c²
7² + 8² = 9²
49 + 64 = 81
113 = 81, no
12, 15, 20
a² + b² = c²
12² + 15² = 20²
144 + 225 = 400
369 = 400, no
Algebra in
Measurement
Finding unknown measurements
Finding measurements
using the coordinate
graph make finding the
area much easier. It’s
just a matter of
counting the number of
units for each length of
the base and the height.
Then all you have to do
is divide by 2.
Take another look at the
previous question and
find the area.
Find the area of the
triangle to the right.
What is your first step to
determine the area.
There are 2 formulas you
must know, what are
they?
Find the area of the rectangle below.
What steps are needed to complete this
problem?
Remember me, the perimeter problem you
needed to find the outside length. Well now
you can find the perimeter using the
Pythagorean Theorem. Find the perimeter and
area of the figure below.
Loading this delivery
truck is very difficult
when you must load
heavy containers, so
the company decided
to build a ramp to
make loading easier.
When the contractor
started to build the
ramp he needed to
know the height of
the back of the ramp.
What do you needed
to do to determine
the height of the
ramp.