1. Three-Dimensional Shapes
One-,
Two-,
Three-Dimensional Shapes
State of Montana Math Standards:
Number Sense and Operation Content Standard 1: 1.4 Common Fractions and Decimals: Identify and model common fractions such as, tenths, fourths, thirds, and halves; and decimals such as money and place value to 0.001; and recognize and compare equivalent representations; 1.5 Length, Time, and Temperature: Select and apply appropriate standard units and tools to measure length, time, and temperature within relevant scientific and cultural situations, including those of Montana American Indians.
Geometric Reasoning Content Standard 3: 3.1 Two-Dimensional Attributes: Describe, compare, and analyze attributes of two-dimensional shapes; 3.2 Three-Dimensional Attributes: Describe attributes of three-dimensional shapes such as cubes and other rectangular prisms, cylinders, cones, and spheres; 3.4 Linear Measurement: Estimate and measure linear attributes of objects in metric units such as centimeters and meters and customary units such as inch, foot, and yard; 3.5 Area and Perimeter: Define and determine area and perimeter of common polygons using concrete tools such as grid paper, objects, or technology and justify the strategy used.
Algebraic and Functional Reasoning Content Standard 4: 4.1 Patterns and Relations: Describe, extend, and make generalizations about geometric or numeric patterns
Duane B. Karlin
CEP 811
June 12, 2011

2. 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.

3. Geometry is the study of shapes.
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.
Ready to try Question 8 again?

4. 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.

5. READY TO LEARN ABOUT… One-dimensional shapes? Two-dimensional shapes?
Three-dimensional shapes?
Or are you ready to TEST YOUR KNOWLEDGE?

6. One-Dimensional Shapes
One-dimensional shapes are measured in only one direction.
This is defined as the LENGTH.
LINES are a one-dimensional shape.

7. One-Dimensional Shapes
One-dimensional shapes are measured in only one direction.
This is defined as the LENGTH.
LINES are a one-dimensional shape.
Ready to try Question 1 again?

8. 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.

9. A Triangle is a “three-sided” shape.
Quadrilateral means “four-sided” shape.
An Octagon is an “eight-sided” shape.
Ready to try Question 9 again?

10. Radius
CIRCLE
Diameter
Center
Circumference

11. CENTER CENTER: the middle of a circle. It is the same distance
from the
center to any
point on the
circle.
Center

12. DIAMETER
Diameter
DIAMETER: a line segment that passes through the center of a circle
and has its endpoints on opposite sides of the circle.

13. Radius
RADIUS
RADIUS: a line segment with one endpoint at the center of a circle
and the other endpoint on the circle.

14. CIRCUMFERENCE CIRCUMFERENCE: the distance around a circle.

15. CIRCUMFERENCE, instead of PERIMETER, is used to
measure the distance around a CIRCLE.
CIRCUMFERENCE = 2πr
π = 3.14
r = radius
3 inches
C = 2 x 3.14 x 3
C = 6.28 x 3
C = 18.84
CIRCUMFERENCE = inches

16. AREA of a CIRCLE is the INTERIOR space.
AREA = πr2
A = 3.14 x 32
3 inches
A = 3.14 x 3 x 3
3 inches
A = 3.14 x 9
A = 28.26
AREA = square inches

17. TRIANGLE The prefix “TRI-” means 3. 3 interior angles 3 sides
INTERIOR means inside.
The sum of the 3 interior angles always equal 180°.

18. AREA of a TRIANGLE = ½ BASE (b) x HEIGHT (h)
A = ½b x h
A = ½ x 6 x 6
A = 3 x 6
A = 18 square inches
HEIGHT
(6 inches)
BASE
This formula works for ALL TRIANGLES.
(6 inches)

19. 6 types of TRIANGLES.
Equilateral
Isosceles
Scalene
Right
Acute
Obtuse
Click on a shape to learn more, or learn about AREA.

20. EQUILATERAL TRIANGLE 60° All three sides are the same length.
All interior angles equal 60°.
(60° + 60° + 60° = 180°)
60°
60°
EQUILATERAL TRIANGLE

21. ISOSCELES 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

22. SCALENE TRIANGLE All three sides are different lengths.
All interior angles are different, but
they still equal 180°.
SCALENE TRIANGLE

23. SCALENE TRIANGLE All three sides are different lengths.
All interior angles are different, but
they still equal 180°.
SCALENE TRIANGLE
Ready to try Question 6 again?

24. RIGHT TRIANGLE One angle, opposite the longest side,
measures 90°. It is signified by the ☐ symbol.
RIGHT TRIANGLE

25. ACUTE 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

26. One interior angle in an obtuse triangle is greater than 90°.

27. QUADRILATERALS
The prefix “QUAD-” means 4, as in a 4-sided figure or shape.
Click on a shape to learn more.

28. PERIMETER = distance around a shape
P = 12 inches
3 inches
3 inches
3 inches
3 inches
PERIMETER of any shape is calculated by adding the sides together.

29. AREA = square units it takes to fill a shape
AREA = 3 x 3
A = 9 square inches 1 2 3
1 inch
1 inch 4 5 6
3 inches
1 inch 7 8 9
3 inches
AREA of a QUADRILATERAL is calculated by multiplying the
Length (or Base) by the Width (or Height).

30. SQUARE All 4 sides are equal and parallel.
All interior angles equal 90°.
REMEMBER:
A square is a
rectangle, but
a rectangle is
not a square!
SQUARE
Parallel means the lines always maintain the same distance apart.
Parallel lines will never touch.

31. SQUARE REMEMBER: A square is a rectangle, but a rectangle is
NOT a square!
SQUARE
Ready to try Question 7 again?

32. RECTANGLE All interior angles equal 90°.
Opposite sides are equal and parallel.

33. RHOMBUS, or DIAMOND Interior angles equal 90°.
A special type of PARALLOGRAM.
All 4 sides are equal and parallel.

34. PARALLELOGRAM Opposite sides are equal and parallel.
Opposite angles are equal.

35. Has one pair of parallel sides.
TRAPEZOID

36. 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 ( ) x 5
A = ½ x (25) x 5
A = 12.5 x 5
AREA = 62.5 square inches

37. 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 ( ) x 5
A = ½ x (25) x 5
A = 12.5 x 5
AREA = 62.5 square inches
Ready to try Question 2 again?

38. 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°.

39. 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°.
Ready to try Question 10 again?

40. SHAPES WITH MORE THAN 4 SIDES
Click on a shape to learn more.

41. PENTAGON The prefix “PENTA-” means 5. 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.

42. PENTAGON The prefix “PENTA-” means 5. No parallel sides.
Ready to try Question 5 again?

43. AREA of a PENTAGON
Divide the pentagon into 5 equal triangles.
A = ½ x 3 x 5
Divide those triangles in half.
A = 1.5 x 5
A = 7.5
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.
BASE = 3 inches
HEIGHT = 5 inches
A = 7.5 x 10
You now have 10 right angle triangles.
AREA = 75 square inches
The formula for finding the
area of a triangle is A = ½ b x h

44. HEXAGON The prefix “HEXA-” means 6. Interior angles all equal 720°.
If each side is equal,
which they do not
have to be, then each
interior angle equals 120°.
3 pairs of parallel sides.
Parallel sides are opposite each other.

45. OCTAGON The prefix “OCTA-” means 8. Interior angles all equal 1080°.
If each side is equal,
which they may or may
not be, then each interior
angle equals 135°.
4 pairs of parallel sides.
Parallel sides are opposite each other.

46. 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.

47. 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.

48. 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.

49. 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.
Ready to try Question 4 again?

50. EDGES The EDGE of a shape is the line where two surfaces meet.
This CYLINDER has
2 EDGES.

51. CUBE The CUBE has 6 sides, 8 vertices, and 12 edges.
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
= 9 x 6
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.

52. CUBE VOLUME is the amount of space a three-dimensional shape occupies.
VOLUME = L x W x H
VOLUME = 4 x 4 x 4
VOLUME = 64 cubic inches
4 inches
HINT: “CUBIC” measurement
is used with volume because
64 equal-sized cubes would
fit into the shape.
4 inches
4 inches
To find the VOLUME of a shape, use this formula: Length x Width x Height.

53. SPHERE DIAMETER = 8 inches, so the RADIUS equals 4 inches.
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 = square inches
Ready to learn about the VOLUME of a SPHERE?

54. 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)
= 4/3 x π x 64
8 inches
= x 64
VOLUME = cubic inches
The RADIUS is half of the DIAMETER, so half of 8 is 4.

55. CYLINDER 2 inches If we cut the middle and lay it flat, it would
form a rectangle.
6 inches
Click on the dotted
line to see what the
cylinder would look
like if it was “dissected.”
A CYLINDER is actually
two circles (one on the
top and one on the bottom)
and a rectangle in the middle.

56. CYLINDER To see the CYLINDER in this shape
makes calculating the SURFACE AREA
easier to understand.
The formula looks
confusing, but it is
simply finding the
surface area of two
circles and one
rectangle.
SURFACE AREA = square inches
6 inches
The circumference
of the circle actually
forms the base of
the rectangle.
SURFACE AREA = 2πr2 + 2πrh
= 2π22 + 2π2 x 6
2 inches
= 2π4 + 2π12
= 6.28 x x 12 =

57. CYLINDER
3 faces
Ready to try Question 3 again?

58. CYLINDER 2 inches To calculate the VOLUME of a CYLINDER, use this
formula: V = πr2h
6 inches
V = π x 22 x 6
V = π x 4 x 6
V = π x 24
V = cubic inches

59. 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.

60. TEST YOUR KNOWLEDGE OF SHAPES
QUESTION 1
How many dimensions does a line have?
One
Two
Three
As many as it needs

61. QUESTION 2
Which of the following formulas would be used to
calculate the area of a trapezoid?
A = ½ B x H
A = L x W
A = ½ (Base 1 + Base 2) x Height
A = πr2

62. How many faces does a cylinder have?
QUESTION 3
How many faces does a cylinder have?
Three
Two
Five
Eight

63. On a three-dimensional shape, what is it called
QUESTION 4
On a three-dimensional shape, what is it called
where two or more points meet?
Face
Vertex
Mystery
Party

64. QUESTION 5
How many parallel sides are on a pentagon? 5 3 2

65. QUESTION 6
Which of these figures is a scalene triangle?

66. QUESTION 7
True or false?
A square is a rectangle and a rectangle is a square.
TRUE
FALSE

67. 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.

68. QUESTION 9
If I had a quadrilateral, two octagons, and a triangle,
how many sides would I have? 19 23 25 15

69. 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

70. EXCELLENT!

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90. Your knowledge of shapes is out of this world!
CONGRATULATIONS!
Your knowledge of shapes is out of this world!
Finished? Return HOME or RAISE YOUR HAND for the teacher!