1. Perimeters, areas and other measurements In many careers it is essential to have the ability to recognize 2-dimensional images as 3-dimensional objects. House builders use 2-dimensional plans to construct a 3-dimensional house.

2. Perimeters and Areas How many measurements would it take to find the length of a wall in the classroom? One This is one-dimensional measurement. (linear) How many measurements would it take to find the size (area) of the floor? Two This is two-dimensional measurement. (length and width)

3. Perimeter is the distance around a geometric figure. Perimeter is a linear measurement, in other words it is one-dimension (length). To find the perimeter P of a polygon, you can add the lengths of its sides.

4. Properties of a rectangle Opposite sides of a rectangle are the same length (congruent). The angles of a rectangle are all congruent (90°). Remember that a 90° angle is called a “right angle.” So, a rectangle has four right angles. Opposite angles of a rectangle are congruent. Opposite sides of a rectangle are parallel.

5. You can find the perimeter of a rectangle by adding the lengths of its sides, Or, since opposite sides of a rectangle are equal length, you can find the perimeter by using the formula: P = 2l + 2w

6. Area: the number of square units in a figure.

7. People talk about the area of all kinds of things. Area is always measured in some size squares. They can be smaller or bigger, depending on the unit of measurement. Here are two different types of measurement, kilometers and inches.

8. Area is the number of square units needed to cover a figure. Here we don’t have any particular unit for measuring the lengths of the sides of the rectangle. The sides are 7 and 4. Can you figure out how many squares is the area? Dimensions are measured in squares so using the proper unit is important. yd 2 cm 2 miles 2

9. Area of a square A = lw or A = S²

10. Area of a rectangle A = lw or A = bh

11. The area A of a rectangle is the product of its length (l) and its width (w) A = lw or A = bh

12. Find the area of a rectangle: Some problems may require you to find an additional piece of information before finding the area. This problem expects you to use the Pythagorean Theorem to find the base of the rectangle before finding the area.

13. Area of a parallelogram A = bh

14. The base of the parallelogram is the length of the rectangle. The height of the parallelogram is the width of the rectangle. The area A of a parallelogram is the product of its base b and its height h. A = bh The height of a parallelogram is not the length of its slanted side. The height of a figure is always perpendicular to the base.

15. Find the area of a parallelogram: When working with parallelogram problems, be sure the height you are using is in fact perpendicular (makes a right angle) to the base (side) you are using. In this problem, 8 is the base and 9 is the height. The side of 10 is not used in this area.

16. Area of a triangle A = ½bh

17. We have discussed how a square, rectangle or parallelogram can be divided by drawing a diagonal from one corner to the opposite corner. This forms two congruent triangles. Finding the area of a triangle from here is fairly simple, just take ½ of the area of the square, rectangle, or parallelogram, So, the area A of a triangle is half the product of its base b and its height h A = ½ bh

18. Find the area of the triangle: It may be necessary, when working with an obtuse triangle, to look outside the triangle to find the height. Notice how the height is drawn to an extension of the base of the triangle. Remember, the height of a triangle or quadrilateral is a line perpendicular from the base to the opposite vertex or side.

19. Some more triangles and their measures. Just a note to remember, the height of a figure is always perpendicular to the base.

20. What do you have to know before you can find the area of the triangle to the left?

21. Use the technique we just discussed and determine the area of the triangle to the left. The length of the height is 4 and the length of the base is (AC) 8. Now multiply (4 · 8) ⁄ 2 = 16 units 2

22. Area of a trapezoid A = ½h(b¹ + b²)

23. A parallelogram can be divided into two congruent trapezoids. The area of each trapezoid is one-half the area of the parallelogram. The two parallel sides of a trapezoid are its bases. If we call the longer side b 1 and the shorter side b 2, then the base of the parallelogram is (b 1 + b 2 ).

24. Find the area of the trapezoid: When working with a trapezoid, the height may be measured anywhere between the two bases. Also, beware of “extra” information. The 35 and 28 are not needed to compute this area.

25. Trapezoid formula:

26. Remember, perimeter is the distance around a figure and is measured in linear units. Area is the space inside a figure and is measured in square units.

27. Composite Figures Several shapes in one

28. Composite shapes offer a unique challenge. They can be several basic shapes together that make up one larger shape.

29. Find the perimeter and area of this figure. Do you have enough information? Sometimes you need to use several different formulas to complete the problem.

30. Find the perimeter and area of the figure.

31. Composite figures drawn on coordinate grids are easily label with dimensions. Draw the figure and find the perimeter and area.

32. Find the perimeter and area of the figure.

34. Pythagorean Theorem Using formulas to find measurements

35. The Egyptians measured their fields with lengths of knotted rope. The size of the farmer’s field is used to work out how big his yield would be, and how much tax he should pay. This knotted rope indicates a triangle with sides of length 3, 4, and 5 units. Pythagoras’s theory is all about right-angled triangles. If squares are constructed on three sides of a right angled triangle, then these squares have a simple but very important connection.

36. Right angles and right- angled triangles have been at the forefront of building since ancient times and is still used today.

37. Pythagorean Theorem - In any right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.

38. Pythagorean Theorem a² + b² = c²

39. What does that mean? It means that if I take leg a which is 3 units and square it (3²) and add it to leg b which is 4 units and square it (4²), that together they will equal the square of the hypotenuse c (5²) a² + b² = c² 3² + 4² = 5² 9 + 16 = 25 25 = 25

40. First, let’s identify the parts of a right triangle. The legs are the two shorter sides of the triangle, and the hypotenuse is the longest side. The hypotenuse is the side that is always opposite the right angle. Did you know? Sailboats have triangular sails to capitalize on wind at 90° to the boat, thereby increasing its maneuverability.

41. Find the length of the hypotenuse. The points form a right triangle with a = 8 and b = 6 a 2 + b 2 = c 2 8 2 + 6 2 = c 2 64 + 36 = c 2 √100 = √c 2 10 = c (0, 9) (6, 1) (0, 1) Remember, the hypotenuse is the side opposite the right angle.

42. Find the length of the hypotenuse. a 2 + b 2 = c 2 8 2 + 15 2 = c 2 64 + 225 = c 2 289 = c 2 √289 = √c 2 17 ≈ c

43. Find the length of leg b. a 2 + b 2 = c 2 6 2 + b 2 = 11 2 36 + b 2 = 121 b 2 = 121 – 36 b 2 = 85 √b 2 = √85 b ≈ 9.219544457…

44. Using Pythagorean Theorem to find area a 2 + b 2 = c 2 a 2 + 8 2 = 12 2 a 2 + 64 = 144 a 2 = 144 – 64 √a 2 = √80 A ≈ 8.94427191 A = ½bh A = ½(16)(8.9) A = 71.2 units 2

45. Find the area and perimeter of the rectangle. a 2 + b 2 = c 2 a 2 + 12 2 = 13 2 a 2 +144 = 169 a 2 = 169 – 144 a 2 = 25 √a 2 = √25 a = 5 A = bh A = (12)(5) A = 60 units 2 P = 2l + 2w P = 2(12) + 2(5) P = 24 + 10 P = 34 units

46. Use the Pythagorean Theorem to find the height of the triangle and the distance across the pond. Could you use the height of the triangle to find the area?

47. The Pythagorean Theorem can be a very useful tool to find measurements.

48. Find the length between the two points, (1, 5), (3, 1).

49. Solving word problems A jogger is taking his normal run for exercise. He leaves home and jogs 8 miles north, then turns and jogs 5 miles west. If he decides to jog straight home, what is the shortest distance he must travel to return to his original starting point.

50. Pythagorean Triples: The distinction between the Pythagorean Theorem and its converse are sometimes over looked. The Pythagorean Theorem states that if a triangle is a right triangle, then the lengths of the sides satisfy the equation a²+ b²= c². The converse says that if you have three numbers that satisfy the equation a²+ b² = c², then those three numbers are side lengths of a right triangle.

51. The most common special sets of triples are below. 3, 4, 5 a²+ b² = c² 3² + 4² = 5² 9 + 16 = 25 25 = 25 5, 12, 13 a² + b² = c² 5² + 12² = 13² 25 + 144 = 169 169 = 169 8, 15, 17 a² + b² = c² 8² + 15² = 17² 64 + 225 = 289 289 = 289

52. Other triples do not work. Remember the three numbers must make the equation a² + b² = c² true. 7, 8, 9 a² + b² = c² 7² + 8² = 9² 49 + 64 = 81 113 = 81, no 12, 15, 20 a² + b² = c² 12² + 15² = 20² 144 + 225 = 400 369 = 400, no

53. Algebra in Measurement Finding unknown measurements

54. Finding measurements using the coordinate graph make finding the area much easier. It’s just a matter of counting the number of units for each length of the base and the height. Then all you have to do is divide by 2. Take another look at the previous question and find the area.

55. Find the area of the triangle to the right. What is your first step to determine the area. There are 2 formulas you must know, what are they?

56. Find the area of the rectangle below. What steps are needed to complete this problem?

57. Remember me, the perimeter problem you needed to find the outside length. Well now you can find the perimeter using the Pythagorean Theorem. Find the perimeter and area of the figure below.

58. Loading this delivery truck is very difficult when you must load heavy containers, so the company decided to build a ramp to make loading easier. When the contractor started to build the ramp he needed to know the height of the back of the ramp. What do you needed to do to determine the height of the ramp.