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**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

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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.

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**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.

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**READY TO LEARN ABOUT… One-dimensional shapes? Two-dimensional shapes?**

Three-dimensional shapes? Or are you ready to TEST YOUR KNOWLEDGE?

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**One-Dimensional Shapes**

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

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**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.

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Radius CIRCLE Diameter Center Circumference

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**CENTER CENTER: the middle of a circle. It is the same distance**

from the center to any point on the circle. Center

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

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

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**CIRCUMFERENCE CIRCUMFERENCE: the distance around a circle.**

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**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

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**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

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**TRIANGLE The prefix “TRI-” means 3. 3 interior angles 3 sides**

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

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**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)

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

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**EQUILATERAL TRIANGLE 60° All three sides are the same length.**

All interior angles equal 60°. (60° + 60° + 60° = 180°) 60° 60° EQUILATERAL TRIANGLE

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**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

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**SCALENE TRIANGLE All three sides are different lengths.**

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

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**RIGHT TRIANGLE One angle, opposite the longest side,**

measures 90°. It is signified by the ☐ symbol. RIGHT TRIANGLE

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**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

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

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

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**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.

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**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).

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**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.

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**RECTANGLE All interior angles equal 90°.**

Opposite sides are equal and parallel.

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**RHOMBUS, or DIAMOND Interior angles equal 90°.**

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

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**PARALLELOGRAM Opposite sides are equal and parallel.**

Opposite angles are equal.

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**Has one pair of parallel sides.**

TRAPEZOID

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**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

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**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°.

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**SHAPES WITH MORE THAN 4 SIDES**

Click on a shape to learn more.

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**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.

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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

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**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.

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**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.

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**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.

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**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.

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**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.

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**EDGES The EDGE of a shape is the line where two surfaces meet.**

This CYLINDER has 2 EDGES.

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**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.

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**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.

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**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?

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**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.

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**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.

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**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 =

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**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

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**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.

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**TEST YOUR KNOWLEDGE OF SHAPES**

QUESTION 1 How many dimensions does a line have? One Two Three As many as it needs

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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

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**How many faces does a cylinder have?**

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

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**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

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

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QUESTION 6 Which of these figures is a scalene triangle?

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

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**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.

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

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**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

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EXCELLENT!

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

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EXCELLENT!

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EXCELLENT!

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EXCELLENT!

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EXCELLENT!

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EXCELLENT!

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EXCELLENT!

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EXCELLENT!

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EXCELLENT!

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EXCELLENT!

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

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**Your knowledge of shapes is out of this world!**

CONGRATULATIONS! Your knowledge of shapes is out of this world!

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