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One-,. Two-,. Three-Dimensional Shapes. Duane B. Karlin. CEP 811. June 12, 2011. 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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  1. One-, Two-, Three-Dimensional Shapes 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. 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.

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

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

  6. 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.

  7. Radius CIRCLE Diameter Center Circumference

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

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

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

  11. CIRCUMFERENCE CIRCUMFERENCE: the distance around a circle. Circumference

  12. 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 = 18.84 inches

  13. 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 = 28.26 square inches

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

  15. AREA of a TRIANGLE = ½ BASE (b) x HEIGHT (h) A = ½b xh 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)

  16. 6 types of TRIANGLES. Equilateral Isosceles Scalene Right Acute Obtuse Click on a shape to learn more, or learn about AREA.

  17. 60° All three sides are the same length. All interior angles equal 60°. (60° + 60° + 60° = 180°) 60° 60° EQUILATERAL TRIANGLE

  18. 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

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

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

  21. 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

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

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

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

  25. 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).

  26. 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.

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

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

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

  30. Has one pair of parallel sides. TRAPEZOID

  31. AREA OF A TRAPEZOID = ½ x (BASE 1 + BASE 2) x HEIGHT 10 inches 5 inches 15 inches Area = ½ x (b1 + b2) xh A = ½ x (15 + 10) x 5 A = ½ x (25) x 5 A = 12.5 x 5 AREA = 62.5 square inches

  32. 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°.

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

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

  35. 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 bxh

  36. The prefix “HEXA-” means 6. HEXAGON 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.

  37. The prefix “OCTA-” means 8. OCTAGON 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.

  38. 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.

  39. 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.

  40. 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.

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

  42. The CUBE has 6 sides, 8 vertices, and 12 edges. CUBE 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.

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

  44. DIAMETER = 8 inches, so the RADIUS equals 4 inches. SPHERE 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 = 200.96 square inches Ready to learn about the VOLUME of a SPHERE?

  45. 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 xπx 64 8 inches = 4.187 x 64 VOLUME = 267.95 cubic inches The RADIUS is half of the DIAMETER, so half of 8 is 4.

  46. 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.

  47. 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 = 100.48 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 4 + 6.28 x 12 = 25.12 + 75.36

  48. CYLINDER 2 inches To calculate the VOLUME of a CYLINDER, use this formula: V = πr2h 6 inches V = πx 22x 6 V = πx 4 x 6 V = πx 24 V = 75.36 cubic inches

  49. 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.

  50. 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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