# Area of a triangle: The area of a triangle can be found with the following formula: You can see why this works with the following diagrams:

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1 Area Review Area of a triangle: The area of a triangle can be found with the following formula: 1 A 2 bh or A bh 2 You can see why this works with the following diagrams: h h b b Solve: Find the area of each triangle in 11in 12in 9in 21in 14in 19in 13in 1 1 Area of a trapezoid: The area of a triangle can be found with the following formula: A 1 2 h( b 1 b 2 ) or A h b 1 b 2 ( 2 ) h b 2 b 1 Solve: Find the area of each trapezoid. 3in in 9in 12in 12in 14in 9in 17in

2 Area Review Area of a circle: The area of a circle can be found with the following formula: Circumference of a circle looks similar: Area and circumference of a circle: Find the area and circumference of each: C 2 r or C 2 A r d 1 32in Combinations: Find the area and perimeter of each: in 1 Review: Find the area and perimeter/circumference of each: in 2 9in 1 12in 2 20in

3 Area Practice Find the area of each: For #7-12 find the circumference/perimeter. ROUND TO THE TENTH WHERE APPLICABLE in 24in 7in 13in 12in 1 14in 2 14in in in 13in 3in 11in 7in 11in 4in in 18n 30in 7. A: C: 8. A: C: 9. A: C: in 7in 11.3in 11in 10. A: P: 11. A: P: 12. A: P:

4 Area of Regular Polygons What formula could be used to determine the area (A) of a regular polygon given the: Number of sides: n Side length: s Apothem (inradius): a A 1 2 asn a s This is easiest to think about as finding the area of n triangles with base s and height a. The area of the pentagon to the right is: A cm 2 10cm How can this formula be simplified given the perimeter P of the polygon? 2 14cm Find the area of each regular polygon below: Round to the tenth. 1. A nonagon (9 sides) whose side length is 12cm and whose apothem is 16.5cm? 2. A hexagon whose sides measure 6 inches and whose apothem is 5.2 inches. 3. An octagon whose sides measure 61 inches and whose apothem is 74 inches. 4. A heptagon whose apothem measure inches and whose sides are 10 inches long. 5. A polygon whose perimeter is 60 inches and whose apothem is 8.5 in?

5 Area of Regular Polygons Determine the area of each figure below: cm 6 cm 5 cm 4 cm What is the perimeter of each figure below? (round to the tenth) 8. Area = 585cm 2 9. Area = 364 in 2 13 cm 10.4 in What is the apothem of each regular polygon below? (round to the tenth) 10. Area = 121 cm Area = 1075 ft 2 5 cm 25 feet

6 Area of Regular Polygons Determine the area of each regular polygon below: cm 9 cm 5 cm 6 cm cm 52cm 8.5 ft 13 ft Given the area, what is the side length of each figure below? (round to the tenth) 5. Area = 120.7in 2 6. Area = 5 2 6cm 4.2 in 5. 6.

7 Polygon Area: Determine the area of each shaded area below: (to the tenth) (all polygons shown are regular) cm 17 feet 2.2cm 2.2cm 14 feet Find the area of the shaded region below: Round to the tenth. 8m 8.2m Challenge: Find the area of the shaded region below: Round to the tenth. 12ft 10.

8 Surface Area Surface Area is the sum of the areas of all faces which enclose a solid. You should alreay be able to find the surface area of basic solids like those below: 4ft 5ft Be methodical! Two ends: 4 x 5 x 2 = 40ft 2 Front and back: 10 x 5 x 2 = 100ft 2 Top and bottom: 10 x 4 x 2 = 80ft 2 10ft Surface area = = 220ft 2 9in 20in Top and bottom = 2(3.14 x 9 2 ) = (remember the formula for area of a circle is Rectangular wrap = 2 x 3.14 x 9 x 20 = in 2 (remember the formula for area of a the wrap is Total surface area: r 2 ( 2 r)h Prisms have identical bases connected by parallelograms (generally rectangles). To find the surface area of a prism, simply add the area of the bases to the area of the lateral faces (sides). Example: (pentagon is regular) 8cm Be methodical! The pentagons are regular: Each pentagon: A = 1/2 x 7 x 10 x 5 = 175cm 2 times 2 = 350 cm 2 7cm 10cm Five lateral faces: A = 8 x 10 = 80 cm 2 times 5 = 400 cm 2 Total surface area = = 750cm 2 Review practice: 1. What is the surface area of a 3-inch tall cylinder with a 7-inch radius? 2. What is the surface area of a 9-foot tall prism whose bases are regular hexagons. Each hexagon has 12-foot sides and a 10-foot apothem.

9 31cm Surface Area Determine the surface area of each solid below: Round all answers to the hundredth. Work on a separate sheet. 3ft 100cm 15ft 3ft 7in 30cm 3. A = 4. A = 5. A = octagon apothem = 12 inches circle diameter = 2ft 9in 3ft 3ft 3ft 6. A = 7. A = 3m Circle radius: 12ft Pentagon sides: 3ft Pentagon apothem: 2ft 20m 10ft 8. A = 9. A =

10 10m 16.5 cm Surface Area: Find the surface area of each: 12m There are 5 surfaces. 9m 30cm There are 9 surfaces. 16cm 10ft 8ft There are 7 surfaces. 15ft cylinder diameter = 18ft diameter = 4ft There are 7 surfaces. 10m 6m

11 31cm Prism/Cylinder 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. This applies whether the figure is right or oblique (Oblique means slanted. Height is measured along the altitude). Practice: Find the volume of each solid. Round to the tenth. 1. (pentagon is regular) 2. 7m 6cm 6m 7cm 10cm cm 12m 13m 5m 10cm 7m circle diameter = 4ft 100cm 30cm 6ft 6ft 6ft 5. 6.

12 Pyramid/Cone Volume The formula used to find the volume of a pyramid or cone: V 1 3 Bh Where B is the area of the base and h is the height. This applies whether the figure is right or oblique (height is measured along the altitude). Practice: Find the volume of each solid. 7. (square-based pyramid) 8. (triangle-based pyramid) ft 10ft 3in 6ft 8ft 10ft 4in 12ft

13 Volume Practice Determine the volume of each solid below: Round all answers to the hundredth. Work on a separate sheet. 15ft 4ft 4ft 3in 4in 8m 15m 17m 1.V = 2. V = 3. V = 4. V = 5. V = 6ft 5ft 5ft 3ft 15ft 5ft 6ft cylinder diameter = 10ft 6. V = 7. V =

14 Review Determine the area of each: Round to the tenth in 40in 50in 20in Determine the area of each: Round to the tenth cm 3.4cm 9cm 21in 5cm 20in Determine the surface area of each: Round to the tenth ft 5ft 7in 11in 7cm 11cm 10cm 10ft cylinder diameter = 12ft Determine the volume of each: Round to the tenth diameter = 4m 9in 17in 1 12in 18m 24m 30m 10m 10m 6m

15 Surface Area and Volume Determine the surface area and volume for each: 3in 7in 11in 4in 4in 9in 1. A = 2. A = 3. A = V = V = V = 11cm 9in 7cm 10cm Small Circle Radius: 4in Large Circle Radius: 4. A = 5. A = V = V = 3in (cylinder with a cone-shaped hole) 3in 4in 6. A = 7. V = V = (no surface area on this one)

16 Surface Area and Volume Determine the surface area and volume for each: diameter = 4ft 13in 7in 10m 12in 17in 10m 6m 1. A = 2. A = 3. A = V = V = V = Surface Area and Volume Determine the surface area and volume for each: diameter = 4ft 13in 7in 10m 12in 17in 10m 6m 1. A = 2. A = 3. A = V = V = V =

17 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? 2x2=4 times bigger. Ex: A cube has one-inch edges. How many times larger is the volume of a cube with edges that are three times longer? 3x3x3=27 times bigger. If you increase the dimensions of an object, the volume increases by the product of those increases. Example: The volume of a rectangular prism is 3. You double the length, width, and height. What will the new volume be? Practice: 1. The area of a reactangle is 15cm 2. If you triple the length and double the width, what will the area of the new rectangle be? 2. A cube has a volume of 2cm 3. Will a cube that has 8 times more volume be twice as tall, three times as tall, 4 times as tall, or 8 times as tall? 3. What happens to the area of a circle when you triple its radius?

18 Changing Dimensions Practice: Solve each. 1. A rectangular prism is 3x4x5 inches. How many times greater is the volume of a 6x8x15 rectangular prism? (If you are not sure, find each volume and divide). 2. When the sides of a triangle are 6 inches long, the area of the triangle is about 15.6 square inches. What would be the area of an equilateral triangle whose sides are 2 inches long? (round to the tenth) 3. A large circle has 81 times the area of a small circle. If the radius of the large circle is 45 inches, what is the radius of the small circle? Practice: Solve each. 1. The radius and height of a cone are tripled. What effect does this have on the cone s volume? 2. The radius of a cylinder is doubled, but the height is not changed. If the original cylinder had a volume of 4cm 3, what is the volume of the new cylinder? 3. A cylinder and a cone have the same base and equal volumes. If the cylinder is 15 inches tall, how tall is the cone? Practice: Solve each. 1. The length and width of a rectangular pyramid are tripled, and the height is doubled. How many times larger is the new pyramid than the original? 2. The dimensions of a cube are increased by 50% (1.5 times). If the original cube had a volume of 1 3, what is the volume of the new cube? 3. You have a square sheet of construction paper. You want a sheet that has twice the area. How many times wider will the new sheet be?

19 Changing Dimensions Complete the following area problems: in 1 30in 4in 24in in 1 Leave answers below in terms of Pi in 30in Complete the following area problems: 10. What happens to the area of a square when you: a. Double the sides. b. Triple the sides. 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. 12. A rectangle has an area of 12cm 2. What will the area be if you: a. Triple all sides. b. Multiply all sides by 1.5.

20 Changing Dimensions Practice: Solve each. 13. A rectangular prism is 2x4x7 inches. How many times greater is the volume of a 6x8x7 rectangular prism? (If you are not sure, find each volume and divide). 14. When the sides of a pentagon are 6 inches long, the area of the pentagon is about 63 square inches. What would the area of a pentagon whose sides are 2 inches long? 15. A large circle has 36 times the area of a small circle. If the radius of the large circle is 24 inches, what is the radius of the small circle? 16. The radius and height of a cylinder are tripled. What effect does this have on the volume? 17. The radius of a cylinder is doubled, and the height is multiplied by 5. If the original cylinder had a volume of 10cm 3, what is the volume of the new cylinder? 18. A right triangle has an area of 2. If all the dimensions are multiplied by 4, what will the area of the new triangle be? 19. The length and width of a rectangular pyramid are doubled, and the height is tripled. How many times larger is the new pyramid than the original? 20. The dimensions of a cube are increased so that they are 2.5 times longer. If the original cube had a volume of 3, what is the volume of the new cube?

21 Changing Dimensions Practice: Solve each. 1. The area of a circle is 30in 2. If you triple the circle s radius, what will its new area be? 2. When a hexagon has 2-inch sides, its area is about 10.4in 2. What will be the approximate area of a hexagon whose sides are 10 inches long?? 3. A rectangular prism has a volume of 17cm 2. 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. 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? a.20% b.23% c.26% d.30% e.40%

22 Area and Volume Practice Test Determine the area of each figure below. Round to the tenth. Figures not to scale in 12in 4in 13in 9in cm 7in Determine the volume for each figure below: (figures not to scale, round to the tenth) 12m 9m 15m 17m 8m 5. Volume = 6. Volume =

23 Area and Volume Practice Test Determine the surface area for each figure below: (figures not to scale, round to the tenth) 7in diameter = 5ft 13in 9in 12in 13in 12m 17in 12m 7m 7. Surface Area = 8. Surface Area = Solve each problem involving changing dimensions: 9. A rectangular prism has a volume of 5cm 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? Pledge and sign: 12.

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