Geometry unit 11.5
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This document provides examples and explanations for calculating the volumes of pyramids, cones, and composite three-dimensional figures. It begins with examples of finding the volumes of rectangular, square, and regular hexagonal pyramids using the volume formula for pyramids. Next, it covers finding the volumes of cones using formulas involving the radius, height, and circumference. The document also explores how changing the dimensions of pyramids and cones affects their volumes. Finally, it presents an example of calculating the total volume of a composite figure made of different shapes.
Geometry unit 11.5
- 1. UNIT 11.5 VOLUMES OFUNIT 11.5 VOLUMES OF PYRAMIDS AND CONESPYRAMIDS AND CONES
- 2. Warm Up Find the volume of each figure. Round to the nearest tenth, if necessary. 1. a square prism with base area 189 ft2 and height 21 ft 2. a regular hexagonal prism with base edge length 24 m and height 10 m 3. a cylinder with diameter 16 in. and height 22 in. 3969 ft3 14,964.9 m3 4423.4 in3
- 3. Learn and apply the formula for the volume of a pyramid. Learn and apply the formula for the volume of a cone. Objectives
- 4. The volume of a pyramid is related to the volume of a prism with the same base and height. The relationship can be verified by dividing a cube into three congruent square pyramids, as shown.
- 5. The square pyramids are congruent, so they have the same volume. The volume of each pyramid is one third the volume of the cube.
- 6. Example 1A: Finding Volumes of Pyramids Find the volume a rectangular pyramid with length 11 m, width 18 m, and height 23 m.
- 7. Example 1B: Finding Volumes of Pyramids Find the volume of the square pyramid with base edge length 9 cm and height 14 cm. The base is a square with a side length of 9 cm, and the height is 14 cm.
- 8. Example 1C: Finding Volumes of Pyramids Find the volume of the regular hexagonal pyramid with height equal to the apothem of the base Step 1 Find the area of the base. Area of a regular polygon Simplify.
- 9. Example 1C Continued Step 2 Use the base area and the height to find the volume. The height is equal to the apothem, . Volume of a pyramid. = 1296 ft3 Find the volume of the regular hexagonal pyramid with height equal to the apothem of the base Simplify.
- 10. Check It Out! Example 1 Find the volume of a regular hexagonal pyramid with a base edge length of 2 cm and a height equal to the area of the base. Step 1 Find the area of the base. Area of a regular polygon Simplify.
- 11. Check It Out! Example 1 Continued Step 2 Use the base area and the height to find the volume. Volume of a pyramid Find the volume of a regular hexagonal pyramid with a base edge length of 2 cm and a height equal to the area of the base. = 36 cm3 Simplify.
- 12. An art gallery is a 6-story square pyramid with base area acre (1 acre = 4840 yd2 , 1 story ≈ 10 ft). Estimate the volume in cubic yards and cubic feet. Example 2: Architecture Application First find the volume in cubic yards. Volume of a pyramid The base is a square with an area of about 2420 yd2 . The base edge length is . The height is about 6(10) = 60 ft or about 20 yd.
- 13. Example 2 Continued Substitute 2420 for B and 20 for h. ≈ 16,133 yd3 ≈ 16,100 yd3 Volume of a pyramid Then convert your answer to find the volume in cubic feet. The volume of one cubic yard is (3 ft)(3 ft)(3 ft) = 27 ft3 . Use the conversion factor to find the volume in cubic feet.
- 14. Check It Out! Example 2 What if…? What would be the volume of the Rainforest Pyramid if the height were doubled? Volume of a pyramid. Substitute 70 for B and 66 for h. = 107,800 yd3 or 107,800(27) = 2,910,600 ft3
- 16. = 245π cm3 ≈ 769.7 cm3 Example 3A: Finding Volumes of Cones Find the volume of a cone with radius 7 cm and height 15 cm. Give your answers both in terms of π and rounded to the nearest tenth. Volume of a pyramid Substitute 7 for r and 15 for h. Simplify.
- 17. Example 3B: Finding Volumes of Cones Find the volume of a cone with base circumference 25π in. and a height 2 in. more than twice the radius. Step 1 Use the circumference to find the radius. Step 2 Use the radius to find the height. h = 2(12.5) + 2 = 27 in. The height is 2 in. more than twice the radius. 2πr = 25π Substitute 25π for the circumference. r = 12.5 Solve for r.
- 18. Example 3B Continued Step 3 Use the radius and height to find the volume. Volume of a pyramid. Substitute 12.5 for r and 27 for h. = 1406.25π in3 ≈ 4417.9 in3 Simplify. Find the volume of a cone with base circumference 25π in. and a height 2 in. more than twice the radius.
- 19. Example 3C: Finding Volumes of Cones Find the volume of a cone. Step 1 Use the Pythagorean Theorem to find the height. 162 + h2 = 342 Pythagorean Theorem h2 = 900 Subtract 162 from both sides. h = 30 Take the square root of both sides.
- 20. Example 3C Continued Step 2 Use the radius and height to find the volume. Volume of a cone Substitute 16 for r and 30 for h. ≈ 2560π cm3 ≈ 8042.5 cm3 Simplify. Find the volume of a cone.
- 21. Check It Out! Example 3 Find the volume of the cone. Volume of a cone Substitute 9 for r and 8 for h. ≈ 216π m3 ≈ 678.6 m3 Simplify.
- 22. Example 4: Exploring Effects of Changing Dimensions original dimensions: radius and height divided by 3: Notice that . If the radius and height are divided by 3, the volume is divided by 33 , or 27. The diameter and height of the cone are divided by 3. Describe the effect on the volume.
- 23. Check It Out! Example 4 original dimensions: radius and height doubled: The volume is multiplied by 8. The radius and height of the cone are doubled. Describe the effect on the volume.
- 24. Example 5: Finding Volumes of Composite Three- Dimensional Figures Find the volume of the composite figure. Round to the nearest tenth. The volume of the upper cone is
- 25. Example 5: Finding Volumes of Composite Three- Dimensional Figures The volume of the cylinder is The volume of the lower cone is The volume of the figure is the sum of the volumes. Find the volume of the composite figure. Round to the nearest tenth. Vcylinder = πr2 h = π(21)2 (35)=15,435π cm3 . V = 5145π + 15,435π + 5,880π = 26,460π ≈ 83,126.5 cm3
- 26. Check It Out! Example 5 Find the volume of the composite figure. The volume of the rectangular prism is V = ℓwh = 25(12)(15) = 4500 ft3 . The volume of the pyramid is The volume of the composite is the rectangular prism subtract the pyramid. 4500 — 1500 = 3000 ft3
- 27. Lesson Quiz: Part I Find the volume of each figure. Round to the nearest tenth, if necessary. 1. a rectangular pyramid with length 25 cm, width 17 cm, and height 21 cm 2. a regular triangular pyramid with base edge length 12 in. and height 10 in. 3. a cone with diameter 22 cm and height 30 cm 4. a cone with base circumference 8π m and a height 5 m more than the radius 2975 cm3 207.8 in3 V ≈ 3801.3 cm3 V ≈ 117.3 m2
- 28. Lesson Quiz: Part II 5. A cone has radius 2 in. and height 7 in. If the radius and height are multiplied by , describe the effect on the volume. 6. Find the volume of the composite figure. Give your answer in terms of π. The volume is multiplied by . 10,800π yd3
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