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One-, Two-, Three-Dimensional Shapes Duane B. Karlin CEP 811 June 12, 2011

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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.

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MEASUREMENTS c an be in U.S. STANDARD o r METRIC. U.S. STANDARD : inches, feet, yards, miles METRIC : meter, decimeter, centimeter, millimeter 12 inches = 1 foot 3 feet = 1 yard 1,760 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.

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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?TEST YOUR KNOWLEDGE?

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

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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). Click on a shape or capital word to learn more. The distance around is PERIMETER. PERIMETER The enclosed space is AREA. AREA Want a hint about INTERIOR ANGLES? INTERIOR ANGLES

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CIRCLE Radius Diameter Circumference Center

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

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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 CIRCUMFERENCE: the distance around a circle.

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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

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AREA of a CIRCLE is the INTERIOR space. AREA = πr 2 3 inches A = 3.14 x 3 2 A = 3.14 x 3 x 3 A = 3.14 x 9 A = 28.26 AREA = 28.26 square inches

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

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

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EquilateralIsoscelesScalene RightAcuteObtuse 6 types of TRIANGLES. Click on a shape to learn more, or learn about AREA.AREA

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

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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.

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

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

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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.

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

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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 AREA = 3 x 3 A = 9 square inches

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

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

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

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

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TRAPEZOID Has one pair of parallel sides.

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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

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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°. 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°.

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

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PENTAGON No 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°.

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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

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

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OCTAGON The prefix “OCTA-” means 8. Interior angles all equal 1080°. 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°.

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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. FACESVERTICES EDGES Click on a shape or capital word to learn more.

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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.

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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! 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.

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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 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.

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SPHERE To find the SURFACE AREA of a sphere, use this formula: SURFACE AREA = 4πr 2 8 inches DIAMETER = 8 inches, so the RADIUS equals 4 inches. = 4π4 2 = 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?

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SPHERE 8 inches To calculate the VOLUME of a SPHERE, things get a little tricky. VOLUME = 4/3 πr 3 = 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.

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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.”

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To see the CYLINDER in this shape makes calculating the SURFACE AREA easier to understand. SURFACE AREA = 2πr 2 + 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. = 2π2 2 + 2π2 x 6 = 2π4 + 2π12 = 6.28 x 4 + 6.28 x 12 = 25.12 + 75.36 SURFACE AREA = 100.48 square inches

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CYLINDER To calculate the VOLUME of a CYLINDER, use this formula: V = πr 2 h 2 inches 6 inches V = π x 2 2 x 6 V = π x 4 x 6 V = π x 24 V = 75.36 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? O NE T WO T HREE A S 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 = πr 2

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QUESTION 3 How many faces does a cylinder have? ThreeTwoFiveEight

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QUESTION 4 On a three-dimensional shape, what is it called where two or more points meet? FaceVertexMysteryParty

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QUESTION 5 How many parallel sides are on a pentagon? 5320

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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. TRUEFALSE

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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? 19232515

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

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