Surface Area of a Cube
• In a cube, all six faces are congruent.
• So, to find the surface area of a cube, we simply
need to find the area of one face, and then multiply it
by 6.
• If we don’t have a cube, but we have a rectangular
prism, there are still 6 faces: but they are not all
congruent.
• Front and back, top and bottom, right and left.
Formulas for surface area
• Cylinder: like a prism with a circular base:
• For prism: 2 • area of the base + perimeter
of the base • height.
• For cylinder: 2 • area of the base +
circumference of the base • height:
• 2 π r2 + 2π rh.
r
r
2πr
h
h
Problem #1
• Suppose you have enough cardboard to make a box
with dimensions
2 inches x 8 inches x 15 inches.
• Vol.: 2 x 8 x 15 = 240 in3.
• SA: 2(2 x 8) + 2(2 x 15) + 2(8 x 15) = 332 in2.
• Find the dimensions of 2 other boxes with the same
surface area. Make up 2 dimensions, say 4 x 10.
Then, solve: 2(4 x 10) + 2(4 x H) + 2(10 x H) = 332.
• One example is 4 x 10 x 9.
Problem #1
• Find the dimensions of 2 other boxes with the same
volume.
• Make up 2 dimensions, say 4 x 10. Then solve: 4 x
10 x H = 240.
• One example is 4 x 10 x 6.
• Write 2 sentences describing your findings.
• Anything along the lines that figures that have the
same surface area may not have the same volume,
and vice versa.
Problem #2
• Sketch a rectangular prism with dimensions a x b x
c. If the area of the base is doubled but the height is
halved, how does the volume change? How does
the surface area change?
• Use real numbers. L = 5, W = 6, H = 14.
• Vol.: 5 x 6 x 14 = 420 un3.
• SA: 2(5 x 6) + 2(5 x 14) + 2(6 x 14) = 368 un2.
• If we double the area of the base, and halve the
height, then: prism might be: 5 x 12 x 7
• Vol: 5 x 12 x 7 = 420 un3 No change.
• SA: 2(5 x 12) + 2(5 x 7) + 2(12 x 7) = 358 un2 Not
equal.
• Sketch a cylinder with radius r and height h. If the
radius is doubled but the height is halved, how does
the volume change? How does the surface area
change?
• Vol: π • r2 • H vs. π • (2r)2 • H/2.
π • r2 • H vs. 4 • 1/2 • π • r2.
• Second cylinder holds twice as much.
• SA: 2 π • r2 + 2 π • r • H vs.
2 π • (2r)2 + 2 π • r • H/2
• Second cylinder has different SA.
• Write 2 sentences describing your findings.
• Anything along the lines of changing these
dimensions does not always yield equal vol. or SA.
Problem #3
• Assume that each block has volume
1 unit3. Make 4 different polyhedra, each containing
12 cubes.
• Sketch or describe each polyhedra. Then, find the
volume and surface area for each.
–
–
–
–
Do all four have the same volume?
Do all four have the same surface area?
Write 2 sentences describing your findings.
Anything along the lines of the volume stays the same but
the SA may change--the more the polyhedron looks like a
rectangular prism, the smaller the SA.
Problem #4
• Without doing any work, predict which cylinder will
have the greater volume and/or greater surface area:
• Cylinder A: radius 6 in., height 6 in.
– Vol.: 216 • π un3.
SA: 144 • π un2.
• Cylinder B: radius 12 in., height 3 in.
– Vol.: 432 • π un3.
SA: 360 • π un2.
• Cylinder C: radius 3 in., height 12 in.
– Vol.: 108 • π un3.
SA: 90 • π un2.
• Check your predictions. Write a sentence
summarizing your findings.
– Anything along the lines that the size of the radius affects
volume and surface area more than the height.
Summary
• Prisms
– Volume: Area of the Base • H of prism
– SA: area of 2 bases + all the faces
• Cylinders
– Volume: Area of Base • H of cylinder = π • r2 • H
– SA: 2 • area of the base + area of the rectangle =
2 • π • r2 + 2 • π • rH
• Pyramids
– Volume: (1/3) • Area of the Base • H of pyramid
– If we had time, we could verify this--for now, see
http://www.wonderhowto.com/how-to/video/how-to-find-thevolume-of-pyramids-in-geometry-181525/
Surface Area
• Literally, the area contained by the
surface of the polyhedron, cone,
cylinder, or sphere.
• Think of it as the amount of paint
needed to paint the outside, or the
exact amount of wrapping paper
needed to wrap the figure.
Find the surface area
• Each pair has a regular, square pyramid and
a triangular prism.
– Step 1: Use wrapping paper, scissors, tape, etc.,
to exactly cover the entire polyhedron. It may be
helpful to trace the faces.
– Step 2: pyramid: Use a ruler to measure the
edge of the base, the height of the pyramid, and
the height of the height of the triangular faces.
– Step 3: prism: Use a ruler to measure the edges
of the triangular base, the height of the triangular
base, and the height of the prism.
– Do not compute! Write out the mathematics you
would need to perform to determine the surface
area of the pyramid and the triangular prism.
For the pyramid…
• Area of the square base…
• Area of 4 triangles (which in this case are all
congruent)…
• So, if the length of the side of the base is b,
the height of the pyramid is H, and the height
of each triangle is s, then the formula is
• b2 + 4 • (1/2 • b • s).
H
b
s
For the triangular prism…
• Area of the triangular bases…
• Area of the rectangular faces…
• If the sides of the triangular base are a, b, c,
and the height of the triangle base is h, and
the height of the prism is H,
• 2 • (1/2 • a • h) + a • H + b • H + c • H =
• 2 • (1/2 • a • h) + (perimeter of the base) • H
c
b
a
h
H
The goal…
• For our purposes in this class and when you
teach this material…
• 1. Help students to understand what all the
variables in the formulas mean.
• 2. Help students to understand where the
formulas come from--help them to develop
the formulas.
• 3. Help students to apply the formulas, and
know when to apply the formulas.
• 4. The goal is not to memorize the
formulas!!!
Sketch, write the formula,
substitute,and compute to find
the surface area
• 1. A cylinder with radius 4 cm and a height
of 8 cm.
• 2. A square pyramid with slant height of 8
in. and the length of the square 10 in.
• A hexagonal pyramid with the area of the
hexagon 40 ft.2 , a slant height of 8 ft., and
the length of the side of the hexagon 6 ft.
• A triangular prism with lengths of the triangle
5 m, 5 m, and 6 m, and the height of the
prism 10 m.
Practice Problem
• Suppose you have a box that measures
length 8 feet x width 10 feet x height 6 feet.
• (a) Find the surface area of this box.
• (b) Suppose you decrease the length by 2
feet and increase the width by 2 feet. Predict
whether the surface area will change or stay
the same. Explain why.
• (c) Now find the new surface area. Can
you explain why your initial prediction was
right or wrong?
Practice Problems
• Suppose you have two similar cubes--one
has side length 4 inches, and the other has
side length 12 inches.
• Find the ratio of the side lengths.
• Find the ratio of the areas of the bases.
• Find the ratio of the surface areas of the
cubes.
• Find the ratio of the volumes of the cubes.
• What is the relationship between the ratios of
the lengths, areas, and volumes?
Make them equal
capacities
• Suppose I have two boxes:
• Where should I make a cut so that the
boxes have the same capacities?
4.5”
8”
8”
2.1”
2.5”
12.5”
Compare the volumes and
surface areas
• A cube with side length 4 inches.
• A cylinder with radius 2 inches and a
height of 4 inches
• In a sentence, explain what you notice.
What percent of the quilt
block is shaded pink?
• Explain your
reasoning.
• If this
pattern is
found on all 6
sides of a cube,
find the percent
of surface area
that is pink.