Surface Area and Surface Area and VolumeVolume

Volume of a Cube• Take a block. Assume that each edge

measures 1 unit.

• So, the volume of that block is 1 unit3. We also call this a cubic unit.

• Use the blocks to make 2 other cubes. How many cubic units are needed?

Volume of a cube• Answer:• 1 cubic unit, 8 cubic units, 27 cubic units• Any “cube” will be formed with x3 blocks. • Ex: a cube with an edge that measures 13

units will have volume of 133, or 2179 cubic units.

Make rectangular prisms• Make 3 different rectangular prisms, each with

a base of 6 cubes.• The base must be a rectangle. Why?• The area of the base remains constant. Why?• The only thing that changes is the height.

Why?• What is the volume (number of cubes) of each

prism? Is this related to the L, W, and H? If so, how?

Dimensions of Rectangular Prisms

• Do your prisms look like this?

• 3 x 2 x 1 3 x 2 x 2

• 3 x 2 x 3 3 x 2 x 4

Rectangular prisms• Volume: Volume is defined as area of

the base multiplied by the height.

• Why do we say L • W • H for a rectangular prism?

length

width

height

Exploration 10.15• Do 3 and 4.

• Show your work and find each answer.

• In 1 - 2 sentences, describe how to imagine the solution to someone who is sight-impaired or blind.

Other 4-sided prisms

• Suppose we had a trapezoidal prism.

• Does the area of the base • height still make sense? (Hint: what is the base?)

3-sided prism• What is the base? What is the height?

Other prisms• Can you find the base and height of

each prism?

What is a prism with a circular base?

• A cylinder.• Does area of the

base • height of the cylinder (prism)still make sense?

• What is area of the base • height?

In prisms and cylinders…• The bases are congruent.

• In a prism, the faces are all rectangles.

• Why aren’t the faces of a cylinder also rectangles?

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.

Volume and Surface Area• Assume that each block has volume

1 unit3. Make 4 different polyhedra, each containing 12 cubes.

• Do all four have the same volume?

• Do all four have the same surface area?

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

hr

h

2πr

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 #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-the-volume-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).

b

H 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

H

h

a

cb

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?

2.5”12.5”

8”

4.5”2.1”

8”

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.