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Geometry

Geometry. Geometry: Part IV Area and Volume By Dick Gill, Julia Arnold and Marcia Tharp for Elementary Algebra Math 03 online. Area.

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Geometry

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  1. Geometry

  2. Geometry: Part IVArea and VolumeByDick Gill, Julia Arnold and Marcia Tharpfor Elementary Algebra Math 03 online

  3. Area

  4. Now that you have completed your module on Perimeter we will turn our attention to Area. Perimeter is the distance around the outside of a figure while the area of the figure is the space contained within the figure. The sketch below for example, represents a square that measures 1 ft. on each side. Its perimeter would be 4 ft. Its area would be 1 square foot or 1 sq. ft. or 1 ft2. 1 ft. The perimeter is the distance around the outside: 4 feet or 4 ft. 1 ft. The area is the space contained inside the figure. Here, the area is 1 square foot or 1 sq.ft or 1 ft2.

  5. 3 ft. 2 ft. If we put two of the squares together, we have a rectangle with a width of 1 ft., a length of 2 ft. and a area of two square feet. 2 ft. 1 ft. If we put six of the squares together as shown in the next sketch, we have a rectangle with a width of 2 ft and a length of 3 ft. and an area of 6 ft2.

  6. 2 ft Here is another rectangle. What is the perimeter? What is the area? By definition, the perimeter is the distance around the outside of the figure. 4 ft 2 ft + 4 ft + 2 ft + 4 ft = 12 ft By formula, P = 2L + 2W = 2(4 ft) + 2(2 ft) = 12 ft. For the area, you can count 8 sq.ft. What would be a formula for the area?

  7. Are you ready to predict the formula for the area of a rectangle? Think before you click. Just multiply the length times the width and you have the area of the rectangle. If the length and width are in meters, the area will be in square meters. If the length and width are in miles, the area will be in square miles. If the length and width are in centimeters, the area will be in square centimeters.

  8. See if you can find the area of each of the following rectangles. The first rectangle has dimensions: L = 5 cm and W = 3 cm. 5 cm A = LW A = (5 cm)(3 cm) A = 15 cm2 or A = 15 sq cm 3 cm OK, how about a rectangle with dimensions: L = 6 in and W = 2 in.? A = LW A = (2 in)(6 in) A = 12 in2 or A = 12 sq in 6 in 2 in

  9. A = LW works for fractional dimensions as well. The rectangle in the sketch has a width of 2 ft. and a length of 2.5 ft. A = LW = (2.5 ft.)(2 ft.) = 5 ft2. If you put the two half squares at the bottom together, you can see how the area would be 5 ft2. Notice how the units behave algebraically. In other words, (ft)(ft) = ft2 just like (x)(x) = x2. 2 ft. 2.5 ft.

  10. A few examples: work out the answer yourself before you click to the solution. Example 1. Find the area of a rectangle with a length of 8 ft. and a width of 3 ft. Solution: A = LW = (8 ft.)(3 ft.) = 24 ft2. Example 2. Find the area of a square 3 ft. on each side. Solution: A = LW = (3 ft.)(3 ft.) = 9 ft2. Since 1 yard is the same as 3 feet does this mean that 1 yd2. = 9 ft2? 1 yd. Yes! 1 yd.

  11. A parallelogram is a four-sided figure whose opposite sides are equal and parallel. Consider the parallelogram in the sketch. The sides on the top and the bottom are called the base sides of the parallelogram and the height is the shortest distance between the two base sides. The height is drawn with a dotted line since it is not one of the sides of the parallelogram. The height will always be perpendicular to the two base sides. In the next few slides we are going to discover the formula for the area of a parallelogram. h b

  12. Note how the height on the left side of the figure creates a triangle. If we move this triangle from the left side of the figure to the right, we will create a rectangle with the same area as the parallelogram. b b b

  13. Note how the height on the left side of the figure creates a triangle. If we move this triangle from the left side of the figure to the right, we will create a rectangle with the same area as the parallelogram. b b h b Since the area of the new rectangle is A = bh, then the area for the parallelogram is A = bh.

  14. So whether your parallelogram is tall or short, start with the base. Then drop a height from the top base to the bottom. h h b b The area of your parallelogram will be A = bh.

  15. Now suppose that you were to cut a parallelogram in half by drawing a diagonal. Since the area of the parallelogram is defined by A = bh, what do you think would be a good formula for the area of the triangle that occupies the lower left half of the parallelogram? If you answered A = ½ bh give yourself a pat on the back. Since each triangle is ½ the original parallelogram, it stands to reason that the area of the triangle would be ½ the area of the parallelogram. h b

  16. Do you think that A = ½ bh would be a good formula for any triangle? This one for example… Does it seem reasonable to you that this triangle is half of some parallelogram? If we flip this triangle

  17. and join the two triangles, we get a parallelogram whose area is defined by A = bh. h The area of the triangle then will be A = ½ bh. b Do you think you can envision every triangle as half of a parallelogram with the same height and base as the triangle?

  18. h h b b h h b b h h b b Before you click, imagine each triangle as half of a parallelogram with the same height and base.

  19. A trapezoid is a four-sided figure with one pair of sides parallel as seen below. The parallel sides are called the bases and the shortest distance from one base of the trapezoid to the other is the height. The height will always be perpendicular to the bases. In the sketch, the longer base is denoted B while the shorter is denoted b. B h b

  20. To discover a formula for the area of a trapezoid, watch what happens when you flip the trapezoid… and fit the two trapezoids together. B b h h B b

  21. To discover a formula for the area of a trapezoid, watch what happens when you flip the trapezoid… and fit the two trapezoids together. You are now looking at a parallelogram whose base is B+b. B b h h b B The area of the parallelogram is A = (B+b)h. Since it took two trapezoids (with the same area) to make the parallelogram the area of one trapezoid is A = ½ (B+b)h

  22. 24 in 14 in 30 in 20m 8m 12m A couple of examples. Work them out before you click through. 1. Find the area of a trapezoid with a height of 14 inches, a base of 24 inches and a second base of 30 inches. A = ½ (B + b)h A = ½ (30in + 24in)14in A = ½ (54in)14in = 378 in2 2. Find the area of a trapezoid with a height of 8 meters, one base of 12 meters and the other of 20 meters. A = ½ (B + b)h A = ½ (20m + 12m)8m A = ½ (32m)8m = 128 m2

  23. We now move our attention to the area of a circle. Thousands of years ago, mathematicians in Egypt and Greece discovered something interesting about a circle—regardless of the size, if you divide the circumference of a circle by its diameter, you always got the same number. Ever since this irrational number has been designated by the Greek letter . Throughout history this number has mystified mathematicians. is an irrational number which means that it cannot be expressed as the fraction of two integers. 22/7 though is a very good approximation. The most frequent approximation is 3.14. If you hit the button on your calculator you will probably see something like this: 3.141592654. We will be expecting you to use your calculator for problems that call for . We also use to calculate the area of circle. Do you know this formula by heart?

  24. r This is the formula that we will use to calculate the area of a circle whose radius is r. We will access from the calculator and then round off. If the radius of the circle above is 5 cm. then the area will be:

  25. Now see what you can do with the following examples. Example 1. Find the area of the circle with a radius of 12 inches. Round your answer to the nearest hundredth. Example 2. Find the area of the circle with a radius of 1 ft. Round your answer to the nearest hundredth. What do the circles in these two examples have in common? Same circle—different units.

  26. Example 3. Find the area of the square with semicircle attached. See if you can do this on your own before you click again. Solution. Since the rectangle is a square, all four sides equal 8m. The top side is the diameter of the semicircle so the radius is 4m. 8m

  27. Practice Problems for Area. Find the area of the indicated figures. (Answers follow.) Round to the nearest hundredth. • Rectangle: L = 152 ft., W = 85 ft. • Square with side = 7.24 mm. • Parallelogram with base of 5.3 cm and height of 4.6 cm. • Rhombus with side = 16.5 in and height = 6.4 in. • Triangle with base = 42 cm and height = 16 cm. • Right Triangle with legs of 18.3 ft and 28.8 ft. • Equilateral triangle with side = 10 ft and height = 8.66 ft. • Equilateral triangle with side = 20 ft. • Circle with radius of 0.478 ft.

  28. Practice Problems for Area Continued. 10. Circle with diameter = 203 mm. 11. Circle with circumference 12.28 m. 12. Semicircle with diameter = 10 in. 13. Quartercircle with radius = 6 in. 14. A square with attached isosceles triangle in the first figure. 15. A rectangle with attached semicircle in the second figure. 5 in 3 in 7 cm 8 in 24 cm

  29. Practice Problems for Area Continued. 16. The rectangle with quarter-circle removed in the first figure. 17. The square with circle removed in the second figure. 16 m 4m 4 m 12 m 18 in

  30. Answers to Practice Problems for Area. • A = LW = (152 ft)(85 ft) = 12,920 sq ft • A = s2 =(7.24 mm)2 = 52.4176 mm2 ~ 52.42 mm2 • A = bh = (5.3 cm)(4.6 cm) = 24.38 cm2 • A = bh = (16.5 in)(6.40 in) = 105.6 in2 • A = ½ bh = ½ (42 cm)(16 cm) = 336 cm2 • A = ½ bh = ½ (18.3 ft)(28.8 ft) = 263.52 ft2 • A = ½ bh = ½ (10 ft)(8.66 ft) = 43.3 ft2 • A = ½ bh = ½ (20 ft)(17.32 ft) = 173.2 ft2 9.

  31. More Answers to Practice Problems for Area.

  32. More Answers to Practice Problems for Area.

  33. Volume By Dr. Julia Arnold

  34. 1

  35. 1 2

  36. 3 2 1

  37. 3 4 2 1

  38. 5 3 4 2 1

  39. 6 5 4 3 2 1

  40. 7 6 5 4 3 1 2

  41. 8 7 6 5 8 Cubes 4 3 1 2

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