Presentation on theme: "Three-Dimensional Shapes"— Presentation transcript:
1 Three-Dimensional Shapes One-,Two-,Three-Dimensional ShapesState 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 patternsDuane B. KarlinCEP 811June 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, miles12 inches = 1 foot3 feet = 1 yard1,760 yards = 1 mileU.S. STANDARD conversions are trickier to memorize because theydo not have a common converting number.METRIC: meter, decimeter, centimeter, millimeter1 meter = 10 decimeters = 100 centimeters = 1,000 millimetersMETRIC conversions are easier to understand becausethey 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?
11 CENTER CENTER: the middle of a circle. It is the same distance from thecenter to anypoint on thecircle.Center
12 DIAMETERDiameterDIAMETER: a line segment that passes through the center of a circleand has its endpoints on opposite sides of the circle.
13 RadiusRADIUSRADIUS: a line segment with one endpoint at the center of a circleand 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 = 2πrπ = 3.14r = radius3 inchesC = 2 x 3.14 x 3C = 6.28 x 3C = 18.84CIRCUMFERENCE = inches
16 AREA of a CIRCLE is the INTERIOR space. AREA = πr2A = 3.14 x 323 inchesA = 3.14 x 3 x 33 inchesA = 3.14 x 9A = 28.26AREA = 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 hA = ½ x 6 x 6A = 3 x 6A = 18 square inchesHEIGHT(6 inches)BASEThis formula works for ALL TRIANGLES.(6 inches)
19 6 types of TRIANGLES.EquilateralIsoscelesScaleneRightAcuteObtuseClick 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 alwaysequal 180° in a triangle.Two sides are equal.The angles opposite ofthe equal sides arealso equal.ISOSCELES TRIANGLE
22 SCALENE TRIANGLE All three sides are different lengths. All interior angles are different, butthey still equal 180°.SCALENE TRIANGLE
23 SCALENE TRIANGLE All three sides are different lengths. All interior angles are different, butthey still equal 180°.SCALENE TRIANGLEReady 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 arean example of anacute triangle, but not allacute triangles areequilateral triangles.ACUTE TRIANGLE
26 One interior angle in an obtuse triangle is greater than 90°.
27 QUADRILATERALSThe 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 inches3 inches3 inches3 inches3 inchesPERIMETER of any shape is calculated by adding the sides together.
29 AREA = square units it takes to fill a shape AREA = 3 x 3A = 9 square inches1231 inch1 inch4563 inches1 inch7893 inchesAREA of a QUADRILATERAL is calculated by multiplying theLength (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 arectangle, buta rectangle isnot a square!SQUAREParallel 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!SQUAREReady 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.
36 AREA OF A TRAPEZOID = ½ x (BASE 1 + BASE 2) x HEIGHT 10 inches5 inches15 inchesArea = ½ x (b1 + b2) x hA = ½ x ( ) x 5A = ½ x (25) x 5A = 12.5 x 5AREA = 62.5 square inches
37 AREA OF A TRAPEZOID = ½ x (BASE 1 + BASE 2) x HEIGHT 10 inches5 inches15 inchesArea = ½ x (b1 + b2) x hA = ½ x ( ) x 5A = ½ x (25) x 5A = 12.5 x 5AREA = 62.5 square inchesReady 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, has180 x (6 – 2) = 180 x 4 = 720°.An OCTAGON, 8-sided shape, has180 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, has180 x (6 – 2) = 180 x 4 = 720°.An OCTAGON, 8-sided shape, has180 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 interiorangle 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 PENTAGONDivide the pentagon into 5 equal triangles.A = ½ x 3 x 5Divide those triangles in half.A = 1.5 x 5A = 7.5But this is only the areafor one triangle, so weneed to multiply thisnumber by the totalnumber of triangleswithin the pentagon.BASE = 3 inchesHEIGHT = 5 inchesA = 7.5 x 10You now have 10 right angle triangles.AREA = 75 square inchesThe formula for finding thearea 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 nothave to be, then eachinterior 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 maynot be, then each interiorangle 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 seeall 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 has2 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), andthen multiply by the total number ofsides (6). Remember to count all thehidden sides!SURFACE AREA = (L x W) x 63 inches= (3 x 3) x 6= 9 x 6SURFACE AREA = 54 square inches3 inches3 inchesSURFACE AREA is the measurementwe would use to cover the outsideof the shape, like a wrapped package.
52 CUBE VOLUME is the amount of space a three-dimensional shape occupies. VOLUME = L x W x HVOLUME = 4 x 4 x 4VOLUME = 64 cubic inches4 inchesHINT: “CUBIC” measurementis used with volume because64 equal-sized cubes wouldfit into the shape.4 inches4 inchesTo 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 ofa sphere, use this formula:SURFACE AREA = 4πr2= 4π428 inches= 4π(4 x 4)= 4π(16)=12.56 x 16SURFACE AREA = square inchesReady 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 648 inches= x 64VOLUME = cubic inchesThe 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 inchesClick on the dottedline to see what thecylinder would looklike if it was “dissected.”A CYLINDER is actuallytwo circles (one on thetop and one on the bottom)and a rectangle in the middle.
56 CYLINDER To see the CYLINDER in this shape makes calculating the SURFACE AREAeasier to understand.The formula looksconfusing, but it issimply finding thesurface area of twocircles and onerectangle.SURFACE AREA = square inches6 inchesThe circumferenceof the circle actuallyforms the base ofthe rectangle.SURFACE AREA = 2πr2 + 2πrh= 2π22 + 2π2 x 62 inches= 2π4 + 2π12= 6.28 x x 12=
58 CYLINDER 2 inches To calculate the VOLUME of a CYLINDER, use this formula: V = πr2h6 inchesV = π x 22 x 6V = π x 4 x 6V = π x 24V = 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 1How many dimensions does a line have?OneTwoThreeAs many as it needs
61 QUESTION 2Which of the following formulas would be used tocalculate the area of a trapezoid?A = ½ B x HA = L x WA = ½ (Base 1 + Base 2) x HeightA = πr2
62 How many faces does a cylinder have? QUESTION 3How many faces does a cylinder have?ThreeTwoFiveEight
63 On a three-dimensional shape, what is it called QUESTION 4On a three-dimensional shape, what is it calledwhere two or more points meet?FaceVertexMysteryParty
64 QUESTION 5How many parallel sides are on a pentagon?532
65 QUESTION 6Which of these figures is a scalene triangle?
66 QUESTION 7True or false?A square is a rectangle and a rectangle is a square.TRUEFALSE
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 9If I had a quadrilateral, two octagons, and a triangle,how many sides would I have?19232515
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 sides2πradd the number of sides together