Squares and Cubes

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Squares and Cubes
Lesson 2.3.6
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Lesson 2.3.6
Squares and Cubes
California Standards Algebra and Functions
1.2 Write and evaluate an algebraic expression
for a given situation, using up to three
variables. Algebra and Functions 3.1 Use
variables in expressions describing geometric
quantities (e.g., P 2w 2l, A  ½bh, C pd
the formulas for the perimeter of a rectangle,
the area of a triangle, and the circumference of
a circle, respectively). Algebra and Functions
3.2 Express in symbolic form simple
relationships arising from geometry.
What it means for you Youll practice using
expressions to represent areas and volumes of
objects.
Key Words

• square
• cube
• edge
• face
• expression
• power
• area
• volume

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Lesson 2.3.6
Squares and Cubes
This is mostly a Lesson on things youve seen
before expressions, equations, order of
operations...
But everything comes about from looking at (and
thinking about) squares and cubes.
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Lesson 2.3.6
Squares and Cubes
Squares Are a Special Sort of Rectangle
Youve already seen that you can write the
formula for the area of a rectangle as A bh.
A
But a square is just a rectangle whose sides all
have the same length call it s.
So the formula for the area becomes
A s s
or A s2
A s s
Remember s 2 (s squared) means exactly the
same as s s.
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Lesson 2.3.6
Squares and Cubes
Example 1
Write an expression for the total area of4
identical squares with sides of length d.
Evaluate your expression in the case where d 3
in.
Solution follows
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Lesson 2.3.6
Squares and Cubes
Example 1
Solution
The area of one square is given by d2.
So the area of 4 identical squares is given by
4d2.
You now have to evaluate this expression for d
3.
The total area will be 4 32 but remember to
evaluate exponents before multiplication.
So the total area is 4 32 4 9 36 in2.
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Lesson 2.3.6
Squares and Cubes
You can use guess and check to find the length
of a side if you know the area.
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Lesson 2.3.6
Squares and Cubes
Example 2
A square has area 36 in2. What is the length of
its sides?
Solution
You need to solve the equation 36 s2. That
means you need to find a value for s that gives
36 when you square it.
Use guess and check

• Try s 10. Here s 2 comes to 10 10 100,
  which is too large.

• Try s 5. Now s 2 comes to 5 5 25, which is
  too small.

• Try s 6. Now s 2 comes to 6 6 36.

So the square must have sides of length 6 inches.
Solution follows
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Lesson 2.3.6
Squares and Cubes
Guided Practice

• Write an expression for the area of a square
  with side length z.
• A square has area 81 cm2. What is the length
  of its sides?
• A square has area 49 in2. What is the length
  of its sides?

z z, or z2
9 cm
7 inches
Solution follows
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Lesson 2.3.6
Squares and Cubes
Guided Practice

• A square has area 16 yd2. What is the length
  of its sides?
• A square has area 64 m2. What is the length
  of its sides?
• A square has area 121 in2. What is the
  length of its sides?

4 yards
8 m
11 inches
Solution follows
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Lesson 2.3.6
Squares and Cubes
Cubes Are Three-Dimensional Squares
A cube is a three-dimensional solid with six
square faces.
Its surface area is the total area of all 6 of
its faces. As each face is a square, the formula
for surface area of a cube is
A 6s2
You might even have to solve equations to do with
cubes.
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Lesson 2.3.6
Squares and Cubes
Example 3
Without using a calculator, find the surface area
of a cube whose edges have length 9 cm.
Solution follows
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Lesson 2.3.6
Squares and Cubes
Example 3
Solution
The surface area of the cube is given by 6s2.
If s 9, this is 6 92
6 81
Calculate the exponent first
6 (80 1)
Rewrite 81 as 80 1
6 80 6 1
Use the distributive property
480 6
Evaluate before
486 cm2
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Lesson 2.3.6
Squares and Cubes
Example 4
The surface area of a cube is 54 in2.
Use guess and check to find the length of the
cubes edges.
Solution
The surface area of the cube is given by the
expression 6s2.
You need to find s such that 6s2 54. Use
guess and check.

• Try s 4. Here 6s2 comes to 6 42 6 16
  96, which is too large.

• Try s 2. Now 6s2 comes to 6 22 6 4 24,
  which is too small.

• Try s 3. Now 6s2 comes to 6 32 6 9 54,
  which is perfect.

So the lengths of the cubes edges must be
3 inches.
Solution follows
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Lesson 2.3.6
Squares and Cubes
Guided Practice

• Write an expression for the surface area of a
  cube with edge length k.
• A cube has surface area 96 in2. What is the
  length of its edges?
• A cube has surface area 216 cm2. What is the
  length of its edges?
• A cube has surface area 150 yd2. What is the
  length of its edges?

6k 2
4 inches
6 cm
5 yards
Solution follows
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Lesson 2.3.6
Squares and Cubes
Volume Is a Measurement of 3-D Space
Volume is a measurement of the amount of space
inside a three-dimensional object.
Its measured in cubic units and equals the
number of unit cubes (cubes whose edges have
length 1) that fit inside the object.
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Lesson 2.3.6
Squares and Cubes
In the diagram on the right, each side has a
length of 2 units, so two unit cubes fit along
each side. (One unit cube is shaded blue.)
You can calculate the volume of a cube using the
formula
or

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Lesson 2.3.6
Squares and Cubes
Example 5
Draw a picture to show the unit cubes in a cube
with edge length 3 cm. Find the volume of the
cube by counting unit cubes. Then verify your
answer using the equation V s 3.
Solution follows
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Lesson 2.3.6
Squares and Cubes
Example 5
Solution
The edges have length 3 cm, so 3 unit cubes
(using units of cm) will fit along each edge.
Count the unit cubes in layers, starting at the
front.
There are 9 unit cubes in the front layer.
And there are 3 layers altogether, which means
there are 9 3 27 unit cubes altogether.
And since each unit cube has a volume of 1 cm3,
the volume must be 27 cm3.
Now use the equation V s3 with s 3 to check
the result
V 33 3 3 3 27 cm3.
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Lesson 2.3.6
Squares and Cubes
Guided Practice

• Using the formula, find the volume of a cube
  with edges 6 meters long.
• By counting the number of unit cubes inside,
  work out the volume of a cube with edges 3
  inches long.
• Use guess and check to find the edge length
  of a cube with a volume of 64 cubic yards.

216 m3
27 in3
4 yards
Solution follows
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Lesson 2.3.6
Squares and Cubes
Independent Practice

• Find the area of a square whose sides have
  length 5 in.
• Find the volume of a cube whose sides have
  length 7 in.

25 in2
343 in3
In Exercise 3, find the value of p.
3.
p in.
p 6
p in.
Area 36 in2
Solution follows
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Lesson 2.3.6
Squares and Cubes
Independent Practice
In Exercises 45, find the value of p.
4.
5.
p in.
p in.
Area 144 in2
Volume p ft3
p 12
p 64
Solution follows
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Lesson 2.3.6
Squares and Cubes
Independent Practice
In Exercises 67, find the value of p.
6.
7.
Edge Length p cm Surface Area 150 cm2
Edge Length p in.Volume 729 in3
p 5
p 9
Solution follows
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Lesson 2.3.6
Squares and Cubes
Round Up
There were a few formulas in this Lesson, but in
the end it always comes down to two skills
being able to substitute values into a formula,
and being able to work backward to find out what
values need to be substituted.
Keep practicing those skills.