Three dimensional geometry

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Three-Dimensional Three-Dimensional GeometryGeometry

Spatial RelationsSpatial Relations

Many jobs in the real-world deal with using three-dimensional figures on two-

dimensional surfaces. A good example of this is architects use drawings to show what the exteriors of buildings will look

Three-dimensional figures have faces, edges, and vertices. A face - is a flat

surface, and edge - is where two faces meet, and a vertex - is where three or more edges

meet. Volume is measured in cubic units.See the example below. Isometric dot paper

can be used to draw three-dimensional figures.

How many faces do most three-dimensional figures

With your isometric dot paper, sketch the drawing below. Make your box 3 units

wide, 2 units high, and 5 units long.

Now try to sketch the box.

After you have sketched the box, try other figures like a cube or pyramid.

Drawing three-dimensional figures

uses a technique called perspective. Here you

make a two-dimensional figure look

like it is three-dimensional.

Let’s try to see if we can draw some three-dimensional figures of

our own.

You will need some isometric dot paper to sketch you drawing.

Next, we are going to make a three-dimensional

figure using lock blocks and then draw our figure and determine how many blocks are used to make

the figure.

3-dimensional objects can also be depicted as

2-dimensional drawings taken at different views.

These representations are called orthogonal drawings.

The 3-dimensional drawing at the left is represented by the 2-dimensional drawings

from the top, front and right-side views.

Volume of Prisms and Volume of Prisms and CylindersCylinders

Measured in cubic unitsMeasured in cubic units33

Volumes of Prisms and Cylinders

A prism is a three-dimensional figure named for the shape of its bases.

Triangular prism has triangles for bases.

Rectangular prism has rectangles for bases.

If all six faces of a prism are squares, it is a cube.

Triangular prism

In this triangular prism the two bases are triangles. The formula for volume of a triangular prism is V = Bh, where B is area of the base and h is height.

Here is another view of a triangular prism. The view on the left shows you how the

prism looks in a 3-dimensional view. The view on the right is the base of the prism.

Find the volume of the prism

V = BhB = area of the base =

area of a triangleV = ½ bh · hV = (.5)(16)(12) = 96 in2

V = Bh height = 12 inV = 96 · 12V = 1152 in3

Volume of the prism is 1152 in3. Volume is measured in cubic units.

Rectangular prism

In this rectangular prism the two bases are rectangles. The volume formula is

V = BhV = (lw)h length · width · height

Find the volume of the prism

V = Bh or V = lwhV = 12 · 8 · 3V = 288 in3

The volume of the prism is 288 in3. Volume is measured in cubic units.

CUBEHere is a 3-dimensional view of a cube.

The view on the left is the cube. The view on the right shows the base of the cube.

The formula for the volume of a cube:V = BhV = lwh

Find the volume of the cube

V = Bh or V =lwhV = 5 · 5 · 5 or 53

V = 125 units3

The volume of the cube is 125 units3. Volume is measured in cubic units.

A die is a cube molded from hard plastic. The edge of a typical die measure 0.62 inches. Dice are

usually produced in a mold which holds 100 die at a time. To the nearest cubic inch, how much plastic is

needed to fill this large mold?

When working with word problems, be sure to read carefully to determine what the question wants you

to find. This question clearly indicates that you are to compute the volume by stating “to the nearest cubic

inch.”

Volume of one die = lwh = (.62)(.62)(.62) = 0.238 cubic inches

For 100 dice = 23.8 = 24 cubic inches

Cylinder: a cylinder is a three-dimensional figure with two circular bases. The volume of a cylinder is the area of the

base B times the height h.

V = Bhor

V = (πr²)h

Find the volume of the cylinder

V = Bh or V = πr2hV = (π · 42) · 10V = 502.4 cm3

The volume of the cylinder is 502.4 cm3. Volume is measured in cubic units.

Effects of Changing Dimensions

By changing the dimensions of a figure, it can have an effect on the volume in different ways, depending on which dimension you

change. Lets look at what happens when you change the dimensions of a prism and a

cylinder.

A juice box measures 3“ by 2“ by 4“. Explain whether doubling the length,

width, or height of the box would double the amount of juice the box holds.

Original V = lwh V = 3·2·4 V = 24 cu.in.

Double length V = lwhV = 6·2·4V = 48 cu.in

Double width V = lwhV = 3·4·4V = 48 cu.in

Double height V = lwhV = 3·2·8V = 48 cu.in.

A juice can has a radius of 1.5 in. and a height of 5 in.. Explain whether doubling

the height of the can would have the same effect on the volume as doubling the radius

Original V = πr²hV = π·1.5²·5V = 11.25π

cu.in.Double V = πr²h radius V = π·3²·5

V = 45π cu.in.Double V = πr²h height V = π·1.5²·10

V = 22.5π cu.in.

Volumes of Pyramids Volumes of Pyramids and Conesand Cones

1/3 of prisms and cylinders1/3 of prisms and cylinders

A pyramid is named for the shape of its base. The base is a polygon, and all the

other faces are triangles.A cone has a circular base.

The height of a pyramid or cone is a perpendicular line measured from the

highest point to the base.

A cone has a circular base. The height of a pyramid or cone is perpendicular line

measured from the highest point to the base.

In the cone to the left the height is h and the radius of the circular base is r.

The s is the slant height which is used to measure surface area of a cone or

pyramid.The volume formula for a cone is

V = 1/3Bh orV = 1/3πr²h

A pyramid is named for its base. The base is a polygon, and all the other faces are

triangles that meet at a common vertex. The height is a perpendicular line from the

base to the highest point.The volume formula for a pyramid is

V = 1/3BhV = 1/3(lw)h

The volumes of cones and pyramids are related to the volumes of cylinders and

prisms.V = πr²h V = Bh

V = 1/3πr²h V = 1/3BhA cone is 1/3 the size of a cylinder with the same base and height. Also, a pyramid is

1/3 the size of a prism with the same height and base.

Finding VolumesFinding VolumesFinding VolumesFinding Volumes

A practical applicationA practical application

Find the volume of the cylinder to the

nearest tenth.

V = BhV = πr2 · h

V = 3.14 · 32 · 8.6V = 243.036 cm3

V = 243 cm3

Find the volume of the prism to the

nearest tenth

V = BhV = 6 · 8 · 2V = 96 cm3

Find the volume of the triangular prism

V = BhV = ½bh · h

V = ½(12 · 16) · 12V = ½(192) · 12

V = ½(2304)V = 1152 in3

Surface Area of Surface Area of Prisms and CylindersPrisms and Cylinders

Back to areasBack to areas22 again again

Surface area of objects are used to advertise, inform, create art, and many other things. On the left is an anamorphic image, which is a distorted picture that becomes recognizable when reflected onto a cylindrical mirror.

One of the most recognizable forms of advertising that uses

surface area of an object is the cereal

box.If you find the

volume, you will find the amount of cereal

the box will hold.If you find the

surface area of the box you determine

how much cardboard is needed to make

the box.

When you flatten-out a three-dimensional object the diagram is called a net. Which of the following answers is the correct net

for the cube. Choose a, b, c, or d.

Finding surface area of figures, for example the box below, can be relatively simple. All is needed is to visualize the faces and then use the appropriate area

formulas for rectangles and circles.

Surface area is the sum of areas of

all surfaces of a figure. The figure

to the left is a rectangular prism. Notice how many

surfaces there are. Lateral surfaces of

a prism are rectangles that

connect the bases.

Top and bottomLeft and right

Front and back

Surface area - is the sum of the areas of all surfaces of a figure. Lateral surfaces -

of a cylinder is the curved surface.

Surface Area: is the number of square units needed to cover all surfaces

of a three-dimensional figure.

Surface area is the sum of the

areas of all surfaces of a

figure. The lateral surfaces of a

triangular prism are triangles and

rectangles.

Two triangular bases and three

rectangles.

Finding Surface AreasFinding Surface AreasFinding Surface AreasFinding Surface Areas

Unfolding the figureUnfolding the figure

Find the surface area of the figure

SA = (top & bottom) + ( front & back)

+ (left & right)

= 2(8 · 6) + 2(8 · 2) + 2(6 · 2)= 96 + 32 + 24SA = 152 cm2

SA = 2(πr2) + lw= 2(area of circle) + (circumference ·

height)= 2(3.14 · 3.12) + (π6.2) · 12

= 60.3508 + 233.616= 293.9668 in2

SA = 2(area of triangle) + (lw) + (lw) + (lw)

= 2(½ · 12 · 16) + (20 · 12) + (16 · 12) + (12 · 12)

= 192 + 240 + 192 + 144= 768 in2

So the next time you see an unusual shape,

just remember geometry is all around

New Year’s Eve ball dropped in New York city each year. The ball is made of 2,668 Waterford crystals with 32,256 LED’s that produce about 16 million different colors.

US Pavilion at the 1967 World Expo in Montreal, Canada.

Three dimensional geometry

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