one-, two-, three-dimensional shapes duane b. karlin cep 811 june 12, 2011

One-,Two-,

Three-Dimensional Shapes

Duane B. Karlin

CEP 811

June 12, 2011

What is DIMENSION?

Dimension is a measure in one direction.

What is GEOMETRY?

Geometry is the study of shapes.

Geometric figures can have one, two, or three dimensions.

MEASUREMENTS can be in U.S. STANDARD or METRIC.

U.S. STANDARD: inches, feet, yards, miles

METRIC: meter, decimeter, centimeter, millimeter

12 inches = 1 foot3 feet = 1 yard1,760 yards = 1 mile

1 meter = 10 decimeters = 100 centimeters = 1,000 millimeters

U.S. STANDARD conversions are trickier to memorize because they do not have a common converting number.

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 are measured in only one direction.

This is defined as the LENGTH.

LINES are a one-dimensional shape.

One-Dimensional Shapes

Two-Dimensional ShapesTwo-dimensional shapes can be measured in two directions.

Their measurements are LENGTH (or BASE) and WIDTH (or HEIGHT).

Click on a shape or capital word to learn more.

The distance around is PERIMETER.

The enclosed space is AREA.

Want a hint about INTERIOR ANGLES?

CIRCLERadius

Diameter

CircumferenceCenter

CENTER

Center

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

DIAMETER

Diameter

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

RADIUSRadius

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 = 2πr

π = 3.14

r = radius

CIRCUMFERENCE, instead of PERIMETER, is used to measure the distance around a CIRCLE.

3 inches

C = 2 x 3.14 x 3

C = 6.28 x 3

C = 18.84

CIRCUMFERENCE = 18.84 inches

AREA of a CIRCLE is the INTERIOR space.

AREA = πr2

3 inches

3 inchesA = 3.14 x 32

A = 3.14 x 3 x 3

A = 3.14 x 9

A = 28.26

AREA = 28.26 square inches

TRIANGLE

3 sides

3 interior angles

The sum of the 3 interior angles always equal 180°.

The prefix “TRI-” means 3.

INTERIOR means inside.

BASE

HEIGHT

AREA of a TRIANGLE = ½ BASE (b) x HEIGHT (h)

A = ½b x h

(6 inches)

(6 inches)

A = ½ x 6 x 6

A = 3 x 6

A = 18 square inches

This formula works for ALL TRIANGLES.

Equilateral Isosceles Scalene

Right Acute Obtuse

6 types of TRIANGLES.

Click on a shape to learn more, or learn about AREA.

EQUILATERAL TRIANGLE

All interior angles equal 60°.

All three sides are the same length.

(60° + 60° + 60° = 180°)

60°

60°60°

ISOSCELES TRIANGLE

Two sides are equal.

The angles opposite of the equal sides are also equal.

REMEMBER: the sum of the interior angles will always equal 180° in a triangle.

SCALENE TRIANGLE

All three sides are different lengths.

All interior angles are different, but they still equal 180°.

RIGHT TRIANGLE

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

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.

OBTUSE TRIANGLE

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

QUADRILATERALS

The prefix “QUAD-” means 4, as in a 4-sided figure or shape.

Click on a shape to learn more.

PERIMETER of any shape is calculated by adding the sides together.

PERIMETER = distance around a shape

3 inches

3 inches

3 inches 3 inches

PERIMETER = 3 + 3 + 3 + 3

P = 12 inches

AREA of a QUADRILATERAL is calculated by multiplying the Length (or Base) by the Width (or Height).

AREA = square units it takes to fill a shape

3 inches

3 inches

AREA = 3 x 3

A = 9 square inches

SQUARE

All 4 sides are equal and parallel.

Parallel means the lines always maintain the same distance apart.Parallel lines will never touch.


REMEMBER: A square is a rectangle, but a rectangle is not a square!

RECTANGLE

Opposite sides are equal and parallel.


RHOMBUS, or DIAMOND

A special type of PARALLOGRAM. All 4 sides are equal and parallel.

Interior angles equal 90°.

PARALLELOGRAM

Opposite sides are equal and parallel.

Opposite angles are equal.

TRAPEZOID

Has one pair of parallel sides.

Area = ½ x (b1 + b2) x h

AREA OF A TRAPEZOID = ½ x (BASE 1 + BASE 2) x HEIGHT

15 inches

10 inches

5 inches

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°.

An OCTAGON, 8-sided shape, has 180 x (8 – 2) = 180 x 6 = 1080°.

A HEXAGON, 6-sided shape, has 180 x (6 – 2) = 180 x 4 = 720°.

SHAPES WITH MORE THAN 4 SIDES

Click on a shape to learn more.

PENTAGONNo parallel sides.

All 5 sides can be equal, but they don’t have to be.

Interior angles all equal 540°.

The prefix “PENTA-” means 5.

If each side is equal, then each interior angle equals 108°.

AREA of a PENTAGON

Divide the pentagon into 5 equal triangles.

Divide those triangles in half.

You now have 10 right angle triangles.

The formula for finding the area of a triangle is A = ½ b x h

A = ½ x 3 x 5

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.

A = 7.5 x 10

AREA = 75 square inches

BASE = 3 inches

HEIGHT = 5 inches

HEXAGON

Parallel sides are opposite each other.

The prefix “HEXA-” means 6.


3 pairs of parallel sides.

If each side is equal, which they do not have to be, then each interior angle equals 120°.

OCTAGON

The prefix “OCTA-” means 8.


4 pairs of parallel sides.

Parallel sides are opposite each other.

If each side is equal, which they may or may not be, then each interior angle equals 135°.

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

FACES refers to the sides of a shape.

In this example, the CUBE has 6 faces, but we can only see 3.

REMEMBER: In a three-dimensional shape, you may not always be able to see all of the faces (sides) of the shape.

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.

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!

3 inches

3 inches

3 inches

SURFACE AREA = (L x W) x 6

= (3 x 3) x 6

= 9 x 6

SURFACE AREA = 54 square inches

SURFACE AREA is the measurement we would use to cover the outside of the shape, like a wrapped package.

CUBE

To find the VOLUME of a shape, use this formula: Length x Width x Height.

VOLUME is the amount of space a three-dimensional shape occupies.

VOLUME = L x W x H

4 inches

4 inches

4 inches

VOLUME = 4 x 4 x 4

VOLUME = 64 cubic inches

HINT: “CUBIC” measurement is used with volume because 64 equal-sized cubes would fit into the shape.

SPHERETo find the SURFACE AREA of a sphere, use this formula:

SURFACE AREA = 4πr2

8 inches

DIAMETER = 8 inches, so the RADIUS equals 4 inches.

= 4π42

= 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

8 inches

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

= 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.”

To see the CYLINDER in this shape makes calculating the SURFACE AREA easier to understand.

SURFACE AREA = 2πr2 + 2πrh

CYLINDER

The formula looks confusing, but it is simply finding the surface area of two circles and one rectangle.

2 inches

6 inches

The circumference of the circle actually forms the base of the rectangle.

= 2π22 + 2π2 x 6

= 2π4 + 2π12

= 6.28 x 4 + 6.28 x 12

= 25.12 + 75.36

SURFACE AREA = 100.48 square inches

CYLINDER

To calculate the VOLUME of a CYLINDER, use this formula: V = πr2h

2 inches

6 inchesV = π x 22 x 6

V = π x 4 x 6

V = π x 24

V = 75.36 cubic 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 = ½ B x H

A = L x W

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 a square.

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!

Oops! Why don’t you try that one again!

EXCELLENT!

CONGRATULATIONS!

Your knowledge of shapes is out of this world!

one-, two-, three-dimensional shapes duane b. karlin cep 811 june 12, 2011

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