# Area is a measure of how much space is occupied by a figure. 1cm 1cm

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1 Area Area is a measure of how much space is occupied by a figure. Area is measured in square units. For example, one square centimeter (cm ) is 1cm wide and 1cm tall. 1cm 1cm A figure s area is the number of square units it would take to cover the figure. It is easy to see that it would take 1 to cover a rectangle that is 3cm by. The area of any rectangle is just the length of its base multiplied by its height. A=bh Practice: Find the area of each shaded region below. Angles are all 90 degrees. 1. 8cm. 1cm 1 3 cm cm cm cm 9cm

2 Area and Perimeter The formula for the area of any parallelogram is the same as the formula for the area of a rectangle: A=bh h = = h b Practice: Find each area. b b 1. 8cm. 3. 1cm 1 cm 3 3cm 7 cm 5 Perimeter is just the distance around a figure. Perimeter is NOT measured in square units. To find the perimeter of a figure, simply add the lengths of its sides. Practice: Find the perimeter and area of each shaded region below. 7cm 1. (square). 10cm 1 4 cm 3cm cm 7cm 1cm 4cm 3cm

3 Area: Triangles Area of a triangle: The area of a triangle can be found with the following formula: A 1 bh or A bh Consider the diagram below to see why the formula works. Practice: Find the area of each triangle in 4in 11in 10in 1in 18in 1in 15in 9in 11in Practice: Find each shaded area: in 3. 4in 16in 16in 4in in in 0in 16in 11in 16in 1in 16in

4 Area: Trapezoids Area of a trapezoid: The area of a trapezoid can be found with the following formula: 1 h( b 1 b ) or h( b 1 b ) Consider the diagram below to see why the formula works. b 1 h b b b 1 Practice: Find the area of each trapezoid in 3. in 4in 8in 3in 8in 13in 10in 3in 10in 8in 8in

5 Perimeter and Area Determine the area and perimeter of each figure below cm 3cm 1 4cm A: P: A: P: cm 13cm 9cm 3cm 9cm 13cm A: P: A: P: cm 0cm 1cm 16cm 1 cm cm A: P: A: P: cm 1cm 10cm 6cm 8cm 17cm 1 9cm 10cm 8cm 17cm A: P: A: P:

6 Circles Area and Circumference of a circle: The Area of a circle can be found with the following formula: Circumference of a circle looks similar: C r or C d A r Area and circumference of a circle: Find the area and circumference of each. Leave your answers in terms of pi in 15in 3in Combinations: Find the area and perimeter of each. Round decimal answers to the tenth in 6in 5in 10in 14.1in 10in 15in Combinations: Find the area and perimeter of each. Round decimal answers to the tenth. 1. 8in. 3. 8in 8in 4in 18in 6in 6in

7 Review Find the area of each figure. Order the areas from least to greatest to spell a question. Answer the question that is created. (Drawings are not to scale). E O A D 16cm 13cm 10cm 1cm 8cm 11cm 8cm 3cm 7cm T N R 1cm T 8cm 9cm 13cm 1cm 1 8cm 9cm 8cm 10cm 1cm I T H A 10cm cm 4cm 6cm 10cm L E A F 1cm 1cm 8cm 1 1cm 14cm 4cm 3cm

8 cm Practice Quiz: Area and Perimeter Find the area and perimeter of each. Include units. All angles that appear to be right angles are right angles cm 1cm 1 9cm 1. Area. Perimeter 10cm cm 10cm 3. Area cm 4. Perimeter cm 5. Area 9cm 6. Perimeter 8cm cm 10cm 10cm 7. Area 8. Perimeter cm 15.6cm 13cm 1cm 1cm 9. Area 10. Perimeter

9 Practice Quiz: Area and Perimeter Answer each. 11. What is the formula for the area of a parallelogram? 1. What is the formula for the area of a circle? 13. What is the formula for the area of a trapezoid? 14. What is the formula for the area of a triangle? 15. What is the formula for the circumference of a circle, using the radius (r)? 11. Area = 1. Area = 13. Area = 14. Area = 15. Circumference = Answer in terms of pi. 6cm 16. Area 17. Circumference 18. Area. Round to the tenth. 10cm 6cm 18. Area Area/Perimeter. Round to the tenth. 6in 6in 6in 19. Area 6in 0. Perimeter

10 Working Backwards: Using Area If we know the area of a figure, we can use it to find the dimensions of a figure. Example: The area of a parallelogram is 4cm and its base is 6cm long. What is its height? 6cm This can be especially useful when finding the altitude (height) of a right triangle. Example: A right triangle has sides of length 6, 8, and 10cm. What is the height (h) of the triangle marked below? h 8cm 10cm h 6cm Given the area of a circle, you can find its radius. Write an equation and solve the following: Example: Approximate the radius of a circle whose area is 6cm. Example: A gallon of paint will cover 400ft. What is the radius of the largest circle that can be painted (and filled-in) with a gallon of paint. Round your answer in feet to the nearest tenth. Challenge: Find the length x in the diagram: x 4cm 7cm 3cm

11 Using Area Practice: Solve each Approximate the side length of a square given each area to the tenth: 1. Area = 16cm.. Area = 90cm. 3. Area = cm. 1. side. side 3. side 4-6. Approximate the radius of each given the area to the tenth: 4. Area = 4 cm. 5. Area = 13cm. 6. Area = 50cm. r 4. radius 5. radius 6. radius 7-9. Find each missing length x. Round decimal answers to the tenth. 7. Area = 14cm. x 4cm 7. x = 8. Area = 4. x cm 7cm 8. x = 8cm 9. Area = 0cm. 9. x = x cm

12 Using Area Practice: Solve each Approximate the height of each rectangle given the area and the base. Fractional answers should be left improper and in simplest form. 1. Area = 80cm. b = 8cm 1. h =. Area = 44cm. b = b h. h = 3. Area = cm. b = 6cm 3. h = 4-6. Approximate the radius of each given the area. Round decimal answers to the tenth. Use 3.14 for pi if you do not have a calculator with a pi button. 4. Area = 9 cm. 5. Area = 30cm. 6. Area = 6.. r 4. radius 5. radius 6. radius 7-9. Find each missing length x. Round decimal answers to the tenth. 7. Area = 140cm. 8cm x 7. x = 8. Area = 84cm. 1cm 7cm x 1cm 8. x = 9. Find x. 1 17cm x 8cm 9. x =

13 Review: Triangles and Trapezoids Area of a triangle: The area of a triangle can be found with the following formula: A 1 bh or A bh Practice: Find the area of each: 1. 7cm 1. Area 4cm. 6. 7cm 1cm. Area 16cm cm Area of a trapezoid: The area of a trapezoid can be found with the following formula: 1 h( b 1 b ) Practice: Find the area of each trapezoid below. or h( b 1 b ) 3. Area 4. 4cm 13cm 4. Area 5. 7cm 4cm 5. Area cm 13cm 1cm 13cm 4cm 6. Area

14 Review: Circles Area and Circumference of a circle: The Area of a circle can be found with the following formula: Circumference of a circle looks similar: C r Practice: Find the area and circumference for each circle described below. Leave answers in terms of pi. or C d A r 7-8. A circle of radius 4cm. 7. Area 8. Circumference A circle of diameter 18cm. 9. Area 10. Circumference A circle of radius 9cm. 11. Area 1. Circumference Practice: Find the area and perimeter/circumference for each circle described below. Round decimal answers to the tenth A circle of radius 10cm. 13. Area 14. Circumference A circle of diameter 7cm. 15. Area 16. Circumference A semi-circle of diameter 4cm 17. Area 4cm 18. Perimeter A quarter-circle of radius 4cm 19. Area 4cm 0. Perimeter

15 Surface Area Surface Area is the sum of the areas of all faces which enclose a solid. Rectangular Prisms are easiest. A rectangular prism has six rectangular faces. To find its surface area, just add the area of all its faces. 10ft 4ft 5ft Rectangular Prism: Two ends: 4 x 5 x = 40ft Front and back: 10 x 5 x = 100ft Top and bottom: 10 x 4 x = 80ft Surface area = = 0ft Other Prisms: Prisms can have bases which are not rectangular. Still, it is easy to just add the area of all the faces of a prism to find its surface area. 6in 10in 8in 7in Triangular Prism: Top and Bottom Triangles: x (6 x 8)/ = 48in Front Right: 8 x 7 = 56in Front Left: 6 x 7 = 4in Back: 10 x 7 = 70in Surface Area: = 16in Practice: Find the surface area of each m 3ft 3ft 4m 4m 15ft 3. 3m 4. 5m 5m 7in 13in 5m 9in 1in 17in 13in 8m

16 4in Surface Area Practice: Find the surface area of each m 8m 4m in 8in 14ft 5in 6ft 3ft 1in in 5in 5in 3in 3in 4in 9in 6in in 6in 5in 13in 4in 1in 13in 8in 16in 3in 10in

17 Surface Area: Cylinders Cut-out the three pieces below the dotted line and use them to build a cylinder. Use this model to write the formula for the surface area of a cylinder, then use the formula to find the surface area of each figure below: 3in 4in 9in 11in 5in 8in r d h d

18 Surface Area: Cylinders Find the surface area of each: 1. A cylinder with a height of 10cm and diameter of. Express your answer in terms of pi.. A cylinder with a height of 4cm and radius of. Express your answer in terms of pi. 3. The lateral surface area of a cylinder does not include the top and bottom. What is the lateral surface area of a cylinder whose height and diameter are each 1cm? 4. The lateral surface area of a certain cylinder is equal to the area of its bases (the circles on top and bottom). If the radius of the cylinder is 4cm, what is its height?

19 Volume Volume: The formula used to find the volume of a prism or cylinder: V Bh Where B is the area of the base and h is the height. 4ft 5ft The base can be any of the six faces. We will use the 10 x 4 side. The volume is the area of the base times the height: (10 x 4) x 5 = 00ft 3. 10ft 9in 0in The base is a circle of area (9) 81. Multiply this by the height to get 0(81 ) 160 in 3. As a decimal, this equals 5,089.4 in 3, but we often leave answers in terms of pi to avoid rounding. Practice: 1. What is the surface area and volume of a 4 by 6 by 7 rectangular prism? SA: in V: in 3. Find the surface area and volume of a 3-inch tall cylinder with a 7-inch radius? SA: in V: in 3 3. What is the surface area and volume of the triangular prism below? 17in 1in SA: in V: in 3 8in 15in

20 Surface Area and Volume Determine the surface area and volume of each. These problems require careful notes. COMPLETE THE WORK ON A SEPARATE SHEET and round all decimal answers to the tenth ft 3cm 15ft 3ft 3cm 3cm 1. SA: Vol.. SA: Vol in 1cm 7in 6in 8in 1 13cm 3. SA: Vol. 4. SA: Vol in 7in 1m 9m note: The portion removed is 1/4 of the total volume. 5. SA: Vol. 6. SA: Vol.

21 Surface Area and Volume Review Determine the surface area and volume of each. Round all decimal answers to the tenth ft 6ft 5in 15in 7in 14ft 4in 1. SA: Vol.. SA: Vol in 7in 10in 5in 4in 5in 4in 9in 3. SA: Vol. 4. SA: Vol.

22 Surface Area and Volume Review Solve each: 5. Find the surface area and volume of a cube whose edges are 4cm long. 5. S.A. Vol. 6. A triangular prism is 3cm tall and has a base whose area is 6cm. What is its volume? 7. A cylinder has a radius of and a height of 6cm. Find its surface area and volume. 6. Vol. 7. S.A. Vol. 8. A circular above-ground swimming pool is 5 feet deep and has a radius of 1 feet. A. How many cubic feet of water will the swimming pool hold? 8A. B. There are about 7.5 gallons in a cubic foot. How many gallons does the pool hold? C. Your garden hose can fill the pool at a rate of 6 gallons per minute. How long will it take to fill the pool with a garden hose? Express your answer to the nearest hour. 8B. D. A rubber liner covers all of the interior walls of the pool as well as the bottom of the swimming pool. What is the total area of the liner? 8C. 8D.

23 Area and Volume Practice Quiz Determine the surface area and volume of each. Round all decimal answers to the tenth in 7in 6in 8in 1. SA:. Vol. 3. SA: 4. Vol in 5in 7in 10in 6in 4in 11in 5. SA: 6. Vol. 7. SA: 8. Vol.

24 Area and Volume Practice Quiz Solve each Find each missing length x. Round decimal answers to the tenth. 9. Area = 14cm. x 4cm 9. x = 10. Area = 30cm. x 10. radius x = 6cm 11. Area = 7cm. 1cm x 18cm 11. x = 1. A right triangle has side lengths of 8cm, 1, and 17cm. What is its area? A circle has a circumference of 10 cm. Express the circle s area in terms of pi The lateral surface area of a cylinder does not include the circles on top and bottom. What is the lateral surface area of a cylinder whose height is 6cm and whose diameter is 8cm? The sides of the square shown are 3cm long, and the radius of the circle is also 3cm. What is the area of the shaded region? Round your decimal answer to the tenth. 15.

25 Pyramid/Cone Volume The formula used to find the volume of a pyramid or cone: V Bh Where B is the area of the base and h is the height. 1 3 Practice: Find the volume of each solid. 1. (square-based pyramid). (triangle-based pyramid) 6in 8in 5in 10in 15in 5in (cone) 4. (cone on top of a cylinder) 4m 30m 3in 5in 18m 6in 4in 3. 4.

26 Volume Practice Determine the volume of each solid below: Round all answers to the hundredth. Work on a separate sheet. 15ft 5ft 5ft 3in 4in 8m 15m 17m 1.V =. V = 3. V = 5in 6in 8in 10in 6in 10in 8in 10in 4. V = (80in is incorrect) 5. V = 6ft 5ft (This is just a cone with the top chopped off... see if you can find a way to get the volume using subtraction.) 8ft 6ft 10ft 15ft 8ft 10ft cylinder diameter = 10ft 1ft 6. V = 7. V =

27 Changing Dimensions Changing the dimensions of an object effects the area and volume. Here are some easy examples: Ex: A square is enlarged so that the length of each side is doubled. If the area of the original square was 7 square inches, what will be the area of the enlarged square? x = 4 times bigger (8in ). Ex: A cube has one-inch edges. How many times larger is the volume of a cube with edges that are three times longer? 3 x 3 x 3 = 7 times bigger. It is easy to see with a square or even a rectangle, but the same concept applies to all shapes and figures. If you increase the dimensions of an object, the area or volume increases by the product of those increases. For example, if you double the base and height of a triangle, how many times greater will the area be? This principle applies to ALL shapes.

28 Changing Dimensions Some more difficult shapes: Circles. What happens when you double the radius of the circle? Does its area double? Try a few examples. Look at the example below. Does the large circle look like it has only twice the area of the smaller circle? When we increase the radius of a circle by a factor x, the area is increased by a factor x. In the formula for area, the radius is squared. Think of this as increasing not only the height, but the height AND width of the circle, or doubling two dimensions. Other shapes. Look at the following regular figures. Each has had its side lengths tripled. What do you think has happened to the area of each figure? Pentagon Area Calculator: cm cm If the area of the figure on the left is 9cm, what is the area of the figure on the right?

29 Changing Dimensions Complete the following area problems: in 5in 6in 10in 18in 30in 4in 8in 4in in 5in 6in 6in 5in 15in Leave answers below in terms of Pi in 6in 30in double the radius 5 times the radius Complete the following area problems: 10. What happens to the area of a square when you: a. Double the side lengths. b. Triple the side lengths. c. Halve the sides. 11. What happens to the volume of a cylinder when you: a. Double the radius only. b. Triple the height only. c. Double the radius and triple the height. 1. A rectangle has an area of 1cm. What will the area be if you: a. Triple all sides. b. Multiply all sides by 1.5.

31 Changing Dimensions Practice: The examples in part a should help you solve each problem b below. 1a. How many times greater is the volume of a cube that has 6-inch sides than a cube that has -inch sides? 1b. A cube has a volume of 3. What would be the volume of a cube whose edge lengths are three times as long? a. One right triangle has sides that are 3, 4, and long. A larger triangle has sides that are 1, 16, and 0cm long. How many times greater is the area of teh large triangle than the small one? b. An equilateral triangle whose sides are 4cm long has an area of about 6.9cm. What is the approximate area of a triangle whose side lengths are 16cm long? 3a. How many times greater is the area of a circle whose radius is 14cm than a circle whose radius is 10cm? 3b. You want to double the area of a circle. Approximately what percent should be added to the radius of the circle? 4a. The square base of a pyramid has sides that are 3cm long. The pyramid is 4cm tall. Another square-based pyramid has 6-inch sides and is also 4cm tall. How many times greater is the volume of the larger pyramid? 4b. A pyramid with a rectangular base has a volume of 0cm 3. If the length and width of the base are doubled, what will be the volume of the new pyramid?

32 Changing Dimensions Practice: Solve each. 1. The area of a circle is 30in. If you triple the circle s radius, what will its new area be?. When a hexagon has -inch sides, its area is about 10.4in. What will be the approximate area of a hexagon whose sides are 10 inches long?? 3. A rectangular prism has a volume of 17cm. If you double the length and width, but leave the height unchanged, what will be the volume of the new prism? 4. If you want to double the area of a square, by what percent should you increase the length of its sides. hint: Try using a 10-inch square, double its area, and find the length of the sides of the new square. Use new/original to approximate the percent. 5. The volume of the regular dodecahedron below with an edge length of 4-inches is about 490 in 3. What would be the volume of a regular dodecahedron whose edges are a foot long? 6. The volume of a cone is 3in 3. What would be the volume after each modification below? (each part refers to the original figure). a. Double the radius only. b. Triple the height only. c. Double the height and triple the radius. d. Increase the height and radius by 50%. 7. If you want to double the volume of a cube, by what percent should you increase the edge length? (Try some examples). a.0% b.3% c.6% d.30% e.40%

33 Changing Dimensions What works for a cube works for any other three-dimensional object. Example: If you double the edge length of a cube, what effect will this have on the: Surface Area? 10cm Volume? Example: If you double the edge length of the shape below (a cuboctahedron), what effect will this have on the surface area and volume? Surface Area? Volume? What if you: Triple the edge lengths? Increase the edge lengths by 40%? 10cm 10cm Practice: 1. The volume of a truncated cube (below) is 10cm 3. What is the volume of a truncated cube whose edge lengths are: a. Twice as long? b. Five times as long? c. 50% longer? (Round to the hundredth.) d. 0% smaller? (Round to the hundredth.). The surface area of the dodecahedron below is 0cm. What is the surface area of a dodecahedron whose edge lengths are: a. Twice as long? b. Five times as long? c. 50% longer? (Round to the tenth.) d. 0% smaller? (Round to the tenth.)

34 Dimensions Practice Quiz Determine the SURFACE AREA of each figure below. Round to the tenth. Figures not to scale. 14ft 5ft 6ft 1. What is the volume of the prism above? 1.. What is the surface area of the prism above? 3. What would the volume be if all three dimensions were doubled?. 5in 3in 3. 3in 4in 4. What is the volume of the prism above? What is the surface area of the prism above? 6. What would the surface area be if all three dimensions were tripled? 5. 3in 4in In terms of pi, what is the surface area of the cylinder above? In terms of pi, what is the volume of the cylinder above? 9. If the radius is doubled and the height remains unchanged, how many times greater will the volume of the new cylinder be? 8. 9.

35 Dimensions Practice Quiz Solve each problem involving changing dimensions: 10. A small pizza has a radius of 10 inches, and a medium pizza has a radius that is 0% larger. How much more pizza do you get with the medium pizza than with the small pizza? Express your answer as a percent A rectangular prism has a volume of 3. If you triple the length, width, and height, what will the volume of the enlarged prism be? When the radius of a circle is multipled by 4, the area of the new circle is 40 in 3. What was the area of the original circle? The volume of a rectangular pyramid is 7m 3. What is the volume of a pyramid that is twice as tall, three times as long, and four times as wide? A cube has edges that are 6 centimeters long. How many times greater is the volume of a cube with 9 centimeter sides? Do not round your answer. 14.

36 Area and Volume Word Problems Many word problems involve comparing the volume between objects. Example: A large cylindrical plastic pitcher is 1 inches tall and has an 8-inch diameter. You use this pitcher to fill a small rectangular fish tank that is 4 inches long and 14 inches wide. How many pitchers will it take to fill the tank to a depth of 16 inches? Example: A palette of bricks is 9 inches wide, 4 inches long, and 7 inches tall. Each brick is 3.65 by.5 by 8. How many bricks are in a palette? Other problems may require that you solve for a missing variable. Example: A 4 r The formula for the surface area of a sphere is of a sphere whose surface area is 36 square inches?. What is the radius Example: The formula for the area of a trapezoid is A h( ). 1 b 1 b If a trapezoid has a height of 8cm, a long base of 1cm, and an area of 76cm, what is the length of its short base? Practice: 1. Cheese is made in large cylindrical shape and then cut into rectangular blocks. About how many 1 by by 4 blocks can be cut from a cheese cylinder that is 18 tall with a 15 radius? (Assume for the problem that no cheese is wasted when the blocks are cut from the cylinder.) Challenge: The formula for the area of an octagon is A s (1 ). If the area of an octagon is 391.1cm, how long are its sides?

37 Area/Volume Word Problems Solve each: 1. A large box contains smaller boxes of tissues. The small tissue boxes are 3 tall, 5 wide, and 9 long. The large box is 7 wide, 5 long, and 4 tall. How many boxes of tissues are in the large box? 1.. The formula for the volume of a cylinder is r h. What is the height of a cylinder whose radius is 6 inches and whose volume is 180 square inches? 3. A ream of paper is 8.5 inches wide, 11 inches long, and 3 inches tall. How many reams are contained within a box that is 17 x 11 x 15? What is the side length of a square whose area is 54cm? Round to the tenth. 5. A manufacturer is shipping boxes in an 18-wheeler. The truck is 40 feet long, 8 feet wide, and 10 feet tall. The boxes are 4 inches wide, 30 inches long, and 1 inches tall. How many boxes will fit in the truck. 6. You want to know how many windows (actually, pieces of glass) are on a building. The building is made entirely of glass, and each piece of glass is 10 feet tall and 8 feet wide. The building is a rectangle 56 feet wide, 80 feet long, and 40 feet tall. How many panes of glass were needed to construct the building, assuming the building is covered entirely in glass (except for the roof) Thomas has an enormous stamp collection. He has wallpapered one wall of his house with stamps that are square with sides that are one and one-quarter inches long. How many stamps were used to cover the wall, which is 10 feet tall and 5 feet long? 7.

38 Area/Volume Word Problems Practice: Solve each. 8. The formula for the area of a hexagon with side length s is a. What is the area of a hexagon of side length 6? 3s 3 A. 8a. b. What is the side length of a hexagon of area 40cm? 8b. 9. Ice cream is sold in a cylinder that is 8 inches tall and has a 3- inch radius. A perfect scoop of ice cream is a perfect sphere with a -inch diameter. The formula for the volume of a sphere is V 4 r 3 a. How many scoops of ice cream are in the container? 3 9a. b. If an ice-cream cone is 3 tall with a 1 radius, how many cones could be filled (just to the top) with the container? 9b. c. Suppose you wanted to fill each cone and add 1/ scoop (a hemisphere) on the top of each. How many ice cream cones couls be made with the container full of ice cream? 9c.

39 Density = Mass/Volume Density is a measure of how much matter is in a given volume. You have probably seen the formula for density in science: d The densities we will use will be in grams per cm 3. Example: A core sample is a cylindrical tube of material cut from the earth. A core sample of stone cut from Mount Mitchell is meters long (00cm), has a radius of 10cm, and a mass of 170 kilograms (440,000 grams). What is its density in grams/cm 3? You can also use a known density to determine the mass of an object. Example: You are building a sand castle at the beach. You fill a bucket of wet sand at the water s edge and carry it to where you are building your castle. Wet sand has a density of 1.9 g/cm 3. a. How much does your bucket weigh in kilograms (1kg = 1,000g) if it is 0cm tall and has a 11cm radius? m v b. There are. pounds in a kilogram. How many pounds does your bucket weigh? Practice: 1. What is the density in g/cm 3 of an object that has a volume of 18cm 3 and a mass of 4g?. What is the density of a cylindrical jar of peanut butter 1cm tall with a 8cm diameter that weighs 748 grams? 3. Granite has a density of.7g/cm 3. You want a granite countertop installed on the island in your kitchen. The countertop is cm thick, 1.3m wide, and m long. How many kilograms will the countertop weigh? (note: 1m = 100cm)

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