1. THE NATURE OF MEASUREMENT Copyright © Cengage Learning. All rights reserved. 9

2. 9.1 Perimeter

3. 3 Measuring Length

4. 4 The most common system of measurement used in the world is what we call the metric system. There have been numerous attempts to make the metric system mandatory in the United States.

5. 5 Measuring Length The most difficult problem in changing from the United States system (the customary system of measurement in the United States) to the metric system is not mathematical, but psychological. Many people fear that changing to the metric system will require complex multiplying and dividing and the use of confusing decimal points.

6. 6 Measuring Length The real advantage of using the metric system is the ease of conversion from one unit of measurement to another. The following box gives the standard units of length.

7. 7 Measuring Length The basic unit of measurement for the U.S. system is the inch. The basic unit of measurement for the metric system is the meter.

8. 8 Measuring Length Greek prefixes kilo-, hecto-, and deka- are used for measurements larger than the basic metric unit, and Latin prefixes deci-, centi-, and milli- are used for smaller quantities (see Figure 9.3). Figure 9.3 Metric prefixes.

9. 9 Measuring Length As you can see from Figure 9.3, a centimeter is of a meter; this means that 1 meter is equal to 100 centimeters.

10. 10 Example 1 – Use a ruler to find a length of segment Measure the given segment a. to the nearest centimeter. b. to the nearest centimeter. Solution: a. b.

11. 11 Perimeter

12. 12 Perimeter One application of both measurement and geometry involves finding the distance around a polygon. This distance is called the perimeter of the polygon.

13. 13 Perimeter Following are some formulas for finding the perimeters of the most common polygons. An equilateral triangle is a triangle with all sides equal in length (see Figure given below) Equilateral triangle PERIMETER = 3(SIDE) P = 3s

14. 14 Perimeter A rectangle is a quadrilateral with opposite sides parallel and equal in length (See the Figure given on right side). A square is a quadrilateral with all sides equal in length (See the Figure given on right side). Rectangle PERIMETER = 2(LENGTH) + 2(WIDTH) P = 2/ + 2w Square PERIMETER = 4(SIDE) P = 4s

15. 15 Example 2 – Find the perimeter Find the perimeter of each polygon.

16. 16 Example 2 – Solution a. Rectangle is 2 cm by 9 cm, so P = 2l + 2w = 2(9) + 2(2) = 18 + 4 = 22 cm

17. 17 Example 2 – Solution b. Rectangle is 2 ft by 4 ft, so P = 2l + 2w = 2(4) + 2(2) = 8 + 4 = 12 ft c. Square, so P = 4s = 4(5) = 20 mi cont’d

18. 18 Example 2 – Solution d. Equilateral triangle, so P = 3s = 3(10) = 30 dm e. Triangle (add lengths of sides), so P = 3 + 4 + 5 = 12 m cont’d

19. 19 Example 3 – Find the length of a pen Suppose you have enough material for 70 ft of fence and want to build a rectangular pen 14 ft wide. What is the length of this pen? Solution: PERIMETER = 2(LENGTH) + 2(WIDTH) 70 = 2(LENGTH) + 2(14) 70 = 2l + 28 Let l = LENGTH OF PEN 42 = 2l 21 = l The pen will be 21 ft long. This is the formula for perimeter. Fill in the given information.

20. 20 Circumference

21. 21 Circumference A circle is the set of all points in a plane a given distance, called the radius, from a given point, called the center. Although a circle is not a polygon, sometimes we need to find the distance around a circle. This distance is called the circumference.

22. 22 Circumference For any circle, if you divide the circumference by the diameter, you will get the same number (see Figure 9.5). Figure 9.5 Circle

23. 23 Circumference This number is given the name  (pi). The number  is an irrational number and is sometimes approximated by 3.14 or We need this number  to state a formula for the circumference C: C = d  where d = DIAMETER or C = 2  r where r = RADIUS

24. 24 Circumference For a circle, the diameter is twice the radius. This means that, if you know the radius and want to find the diameter, you simply multiply by 2. If you know the diameter and want to find the radius, divide by 2.

25. 25 Example 4 – Find distance around a figure Find the distance around each figure.

26. 26 Example 4 – Solution a. C = 4  ; a decimal approximation is C  4(3.14) = 12.56. The circumference is 4  ft, which is about 13 ft. b. C = 28  ; this is about 88 cm. c. C = 2  (5) = 10  ; this is about 31 m. d. This is half of a circle (called a semicircle); thus, the curved part is half of the circumference (C = 2   6.28), or about 3.14, which is added to the diameter: 3.14 + 2 = 5.14 The distance around the figure is about 5 dm.

27. 27 Example 4 – Solution e. This is one-eighth of a circle. The curved part is one eighth of the circumference [C = 2  (30)  188.5],or about 23.6, which is added here to the radius (on both slides): 23.6 + 30 + 30  84 cm cont’d