Transcript One-, Two-, Three-Dimensional Shapes

One-,
Two-,
Three-Dimensional S
Duane B. Karlin
CEP 811
June 12, 2011
What is GEOMETRY?
Geometry is the study of shapes.
Geometric figures can have one, two, or three dimensions.
What is DIMENSION?
Dimension is a measure in one direction.
MEASUREMENTS can be in U.S. STANDARD or METRIC.
U.S. STANDARD: inches, feet, yards, miles
12 inches = 1 foot
3 feet = 1 yard
1,760 yards = 1 mile
U.S. STANDARD conversions are trickier to memorize because they
do not have a common converting number.
METRIC: meter, decimeter, centimeter, millimeter
1 meter = 10 decimeters = 100 centimeters = 1,000 millimeters
METRIC conversions are easier to understand because
they are multiples of 10.
READY TO LEARN ABOUT…
One-dimensional shapes?
Two-dimensional shapes?
Three-dimensional shapes?
Or are you ready to TEST YOUR KNOWLEDGE?
One-Dimensional Shapes
One-dimensional shapes are measured in only one direction.
This is defined as the LENGTH.
LINES are a one-dimensional shape.
Two-Dimensional Shapes
Two-dimensional shapes can be measured in two directions.
Their measurements are LENGTH (or BASE) and WIDTH (or HEIGHT).
The distance around is PERIMETER.
The enclosed space is AREA.
Want a hint about INTERIOR ANGLES?
Click on a shape or capital word to learn more.
CIRCLE
Radius
Diameter
Center
Circumference
CENTER
CENTER: the
middle of a
circle. It is the
same distance
from the
center to any
point on the
circle.
Center
DIAMETER
Diameter
DIAMETER: a line segment that passes through the center of a circle
and has its endpoints on opposite sides of the circle.
RADIUS
Radius
RADIUS: a line segment with one endpoint at the center of a circle
and the other endpoint on the circle.
CIRCUMFERENCE
Circumference
CIRCUMFERENCE:
the distance around
a circle.
CIRCUMFERENCE, instead of PERIMETER, is used to
measure the distance around a CIRCLE.
CIRCUMFERENCE = 2πr
π = 3.14
r = radius
C = 2 x 3.14 x 3
C = 6.28 x 3
C = 18.84
CIRCUMFERENCE = 18.84 inches
3 inches
AREA of a CIRCLE is the INTERIOR space.
AREA = πr2
A = 3.14 x 32
A = 3.14 x 3 x 3
A = 3.14 x 9
A = 28.26
AREA = 28.26 square inches
3 inches
3 inches
TRIANGLE
The prefix “TRI-” means 3.
3 interior angles
3 sides
INTERIOR means inside.
The sum of the 3 interior angles always equal 180°.
AREA of a TRIANGLE = ½ BASE (b) x HEIGHT (h)
A = ½b x h
A=½x6x6
A=3x6
A = 18 square inches
HEIGHT
(6 inches)
BASE
(6 inches)
This formula works for ALL TRIANGLES.
6 types of TRIANGLES.
Equilateral
Right
Isosceles
Acute
Scalene
Obtuse
Click on a shape to learn more, or learn about AREA.
All three sides are the same length.
60°
All interior angles equal 60°.
(60° + 60° + 60° = 180°)
60°
60°
EQUILATERAL TRIANGLE
REMEMBER: the sum of the
interior angles will always
equal 180° in a triangle.
Two sides are equal.
The angles opposite of
the equal sides are
also equal.
ISOSCELES TRIANGLE
All three sides are different lengths.
All interior angles are different, but
they still equal 180°.
SCALENE TRIANGLE
One angle, opposite the longest side,
measures 90°. It is signified by the ☐ symbol.
RIGHT TRIANGLE
All 3 interior angles are
less than 90°.
Equilateral triangles are
an example of an
acute triangle, but not all
acute triangles are
equilateral triangles.
ACUTE TRIANGLE
One interior angle in an obtuse triangle is greater than 90°.
OBTUSE TRIANGLE
QUADRILATERALS
The prefix “QUAD-” means 4, as in a 4-sided figure or shape.
Click on a shape to learn more.
PERIMETER = distance around a shape
PERIMETER = 3 + 3 + 3 + 3
P = 12 inches
3 inches
3 inches
3 inches
3 inches
PERIMETER of any shape is calculated by adding the sides together.
AREA = square units it takes to fill a shape
AREA = 3 x 3
A = 9 square inches
3 inches
3 inches
AREA of a QUADRILATERAL is calculated by multiplying the
Length (or Base) by the Width (or Height).
All 4 sides are equal and parallel.
SQUARE
All interior angles equal 90°.
REMEMBER:
A square is a
rectangle, but
a rectangle is
not a square!
Parallel means the lines always maintain the same distance apart.
Parallel lines will never touch.
All interior angles equal 90°.
RECTANGLE
Opposite sides are equal and parallel.
RHOMBUS, or DIAMOND
Interior angles equal 90°.
A special type of PARALLOGRAM.
All 4 sides are equal and parallel.
Opposite sides are equal and parallel.
PARALLELOGRAM
Opposite angles are equal.
Has one pair of parallel sides.
TRAPEZOID
AREA OF A TRAPEZOID = ½ x (BASE 1 + BASE 2) x HEIGHT
10 inches
5 inches
15 inches
Area = ½ x (b1 + b2) x h
A = ½ x (15 + 10) x 5
A = ½ x (25) x 5
A = 12.5 x 5
AREA = 62.5 square inches
HINT!
Remember, the number of degrees in any geometric shape is
180 x (N – 2), where “N” is equal to the number of sides.
So, with a PENTAGON, 5-sided shape, we would write:
180 x (5 – 2) = 180 x 3 = 540,
so the number of degrees in a PENTAGON is 540°.
A HEXAGON, 6-sided shape, has
180 x (6 – 2) = 180 x 4 = 720°.
An OCTAGON, 8-sided shape, has
180 x (8 – 2) = 180 x 6 = 1080°.
SHAPES WITH MORE THAN 4 SIDES
Click on a shape to learn more.
The prefix “PENTA-” means 5.
PENTAGON
No parallel sides.
If each side is equal,
then each interior
angle equals 108°.
Interior angles all equal 540°.
All 5 sides can be equal, but they don’t have to be.
AREA of a PENTAGON
Divide the pentagon into 5 equal triangles.
A=½x3x5
Divide those triangles in half.
A = 1.5 x 5
A = 7.5
BASE = 3 inches
HEIGHT = 5 inches
But this is only the area
for one triangle, so we
need to multiply this
number by the total
number of triangles
within the pentagon.
A = 7.5 x 10
You now have 10 right angle triangles.
The formula for finding the
area of a triangle is A = ½ b x h
AREA = 75 square inches
The prefix “HEXA-” means 6.
HEXAGON
If each side is equal,
which they do not
have to be, then each
interior angle equals 120°.
Interior angles all equal 720°.
3 pairs of parallel sides.
Parallel sides are opposite each other.
The prefix “OCTA-” means 8.
OCTAGON
If each side is equal,
which they may or may
not be, then each interior
angle equals 135°.
Interior angles all equal 1080°.
4 pairs of parallel sides.
Parallel sides are opposite each other.
Three-Dimensional Shapes
Three-dimensional shapes are measured in three directions:
length, width, and height.
Three-dimensional shapes also have FACES, VERTICES, and EDGES.
Click on a shape or capital word to learn more.
FACES
REMEMBER:
In a three-dimensional
shape, you may not
always be able to see
all of the faces (sides)
of the shape.
FACES refers to the sides of a shape.
In this example, the CUBE has 6 faces, but we can only see 3.
VERTEX (singular), or VERTICES (plural)
A VERTEX is where two or more points meet; a corner.
This example of a RECTANGULAR PRISM has 8 VERTICES.
Once again, not every VERTEX may be visible in a three-dimensional shape.
EDGES
The EDGE of a shape
is the line where two
surfaces meet.
This CYLINDER has
2 EDGES.
The CUBE has 6 sides, 8 vertices, and 12 edges.
CUBE
To find the SURFACE AREA of a CUBE,
find the area of one side (L x W), and
then multiply by the total number of
sides (6). Remember to count all the
hidden sides!
SURFACE AREA = (L x W) x 6
3 inches
= (3 x 3) x 6
=9x6
SURFACE AREA = 54 square inches
3 inches
3 inches
SURFACE AREA is the measurement
we would use to cover the outside
of the shape, like a wrapped package.
VOLUME is the amount of space a
three-dimensional shape occupies.
CUBE
VOLUME = L x W x H
VOLUME = 4 x 4 x 4
VOLUME = 64 cubic inches
4 inches
4 inches
4 inches
HINT: “CUBIC” measurement
is used with volume because
64 equal-sized cubes would
fit into the shape.
To find the VOLUME of a shape, use this formula: Length x Width x Height.
DIAMETER = 8 inches, so the RADIUS equals 4 inches.
SPHERE
To find the SURFACE AREA of
a sphere, use this formula:
SURFACE AREA = 4πr2
= 4π42
8 inches
= 4π(4 x 4)
= 4π(16)
=12.56 x 16
SURFACE AREA = 200.96 square inches
Ready to learn about the VOLUME of a SPHERE?
SPHERE
To calculate the VOLUME of a SPHERE,
things get a little tricky.
VOLUME = 4/3 πr3
= 4/3 π (4 x 4 x 4)
8 inches
= 4/3 x π x 64
= 4.187 x 64
VOLUME = 267.95 cubic inches
The RADIUS is half of the DIAMETER, so half of 8 is 4.
CYLINDER
2 inches
6 inches
A CYLINDER is actually
two circles (one on the
top and one on the bottom)
and a rectangle in the middle.
If we cut the middle
and lay it flat, it would
form a rectangle.
Click on the dotted
line to see what the
cylinder would look
like if it was “dissected.”
CYLINDER
The formula looks
confusing, but it is
simply finding the
surface area of two
circles and one
rectangle.
The circumference
of the circle actually
forms the base of
the rectangle.
To see the CYLINDER in this shape
makes calculating the SURFACE AREA
easier to understand.
SURFACE AREA = 100.48 square inches
6 inches
SURFACE AREA = 2πr2 + 2πrh
2 inches
= 2π22 + 2π2 x 6
= 2π4 + 2π12
= 6.28 x 4 + 6.28 x 12
= 25.12 + 75.36
CYLINDER
2 inches
To calculate the VOLUME
of a CYLINDER, use this
formula: V = πr2h
V=πx
22
x6
V=πx4x6
V = π x 24
V = 75.36 cubic inches
6 inches
RECTANGULAR PRISM
The RECTANGULAR PRISM has
6 sides, 8 vertices, and 12 faces.
To calculate the SURFACE AREA or VOLUME or the RECTANGULAR PRISM,
use the same formula as you would for the CUBE.
TEST YOUR KNOWLEDGE OF SHAPES
QUESTION 1
How many dimensions does a line have?
ONE
TWO
THREE
AS MANY AS IT NEEDS
QUESTION 2
Which of the following formulas would be used to
calculate the area of a trapezoid?
A=½BxH
A=LxW
A = ½ (Base 1 + Base 2) x Height
A = πr2
QUESTION 3
How many faces does a cylinder have?
Three
Two
Five
Eight
QUESTION 4
On a three-dimensional shape, what is it called
where two or more points meet?
Face
Vertex
Mystery
Party
QUESTION 5
How many parallel sides are on a pentagon?
5
3
2
0
QUESTION 6
Which of these figures is a scalene triangle?
QUESTION 7
True or false?
A square is a rectangle and a rectangle is
TRUE
FALSE
QUESTION 8
What is geometry?
The study of numbers.
The study of shapes.
An example of counting.
What the acorn said when it grew up.
QUESTION 9
If I had a quadrilateral, two octagons, and a triangle,
how many sides would I have?
19
23
25
15
QUESTION 10
WHICH FORMULA WILL HELP ME FIGURE OUT HOW MANY DEGREES
ARE IN ANY GIVEN GEOMETRIC SHAPE?
180 x (number of sides - 2)
½ Base x Height x the number of sides
2πr
add the number of sides together
EXCELLENT!
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CONGRATULATIONS!
Your knowledge of shapes is out of this world!