11.5 Volumes of Prisms and Cylinders answers

Presentation on theme: "11.5 Volume of Prisms and Cylinders"— Presentation transcript:

1 11.5 Volume of Prisms and Cylinders
Geometry

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Presentation on theme: "11.5 Volume of Prisms and Cylinders"— Presentation transcript:
How do you find the volume of prisms and cylinders?
How do you find the volume of prisms?
What is the formula for volume of cylinders?

2 Geometry 11.5 Volume of Prisms and Cylinders
Goals Find the volume of prisms. Find the volume of cylinders. Solve problems using volume. Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

3 Geometry 11.5 Volume of Prisms and Cylinders
The number of cubic units contained in a solid. Measured in cubic units. Basic Formula: V = Bh B = area of the base, h = height Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

4 Geometry 11.5 Volume of Prisms and Cylinders
Cubic Unit V = s3 V = 1 cu. unit s 1 1 s 1 s Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

5 Geometry 11.5 Volume of Prisms and Cylinders
Which stack has the largest Volume? Cavalieri’s Principle If two solids have the same height and the same cross-sectional area at every level, then they have the same volume. Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

6 Geometry 11.5 Volume of Prisms and Cylinders
Prism: V = Bh B B h h h B Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

7 Geometry 11.5 Volume of Prisms and Cylinders
Cylinder: V = r2h r B h h V = Bh Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

8 Example 1 Find the volume.
Triangular Prism V = Bh Base = 40 V = 40(3) = 120 10 8 3 Abase = ½ (10)(8) = 40 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

9 Example 2 Find the volume.
V = Bh The base is a ? Hexagon 12 10 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

10 Geometry 11.5 Volume of Prisms and Cylinders
Example 2 Solution 12 12 12 ? ? ? 6 10 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

11 Geometry 11.5 Volume of Prisms and Cylinders
Example 2 Solution V = Bh V = (374.1)(10) V  3741 12 374.1 10 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

12 Geometry 11.5 Volume of Prisms and Cylinders
Example 3 A soda can measures 4.5 inches high and the diameter is 2.5 inches. Find the approximate volume. V = r2h V = (1.252)(4.5) V  22 in3 (The diameter is 2.5 in. The radius is 2.5 ÷ 2 inches.) Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

13 Geometry 11.5 Volume of Prisms and Cylinders
Example 4 A wedding cake has three layers. The top cake has a diameter of 8 inches, and is 3 inches deep. The middle cake is 12 inches in diameter, and is 4 inches deep. The bottom cake is 14 inches in diameter and is 6 inches deep. Find the volume of the entire cake, ignoring the icing. Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

14 Geometry 11.5 Volume of Prisms and Cylinders
Example 4 Solution VTop = (42)(3) = 48  in3 VMid = (62)(4) = 144  in3 VBot = (72)(6) = 294  in3 r = 4 8 3 r = 6 12 4 486  in3 14 6 r = 7 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

15 Geometry 11.5 Volume of Prisms and Cylinders
Concrete Pipe Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

16 Geometry 11.5 Volume of Prisms and Cylinders
Example 5 A manufacturer of concrete sewer pipe makes a pipe segment that has an outside diameter (o.d.) of 48 inches, an inside diameter (i.d.) of 44 inches, and a length of 52 inches. Determine the volume of concrete needed to make one pipe segment. 48 44 52 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

17 Geometry 11.5 Volume of Prisms and Cylinders
Example 5 Solution Strategy: Find the area of the ring at the top, which is the area of the base, B, and multiply by the height. View of the Base Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

18 Geometry 11.5 Volume of Prisms and Cylinders
Example 5 Solution Strategy: Find the area of the ring at the top, which is the area of the base, B, and multiply by the height. Area of Outer Circle: Aout = (242) = 576 Area of Inner Circle: Ain = (222) = 484 Area of Base (Ring): ABase = 576 - 484 = 92 48 44 52 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

19 Geometry 11.5 Volume of Prisms and Cylinders
Example 5 Solution V = Bh ABase = B = 92 V = (92)(52) V = 4784 V  15,029.4 in3 48 44 52 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

20 Example 5 Alternate Solution
48 Vouter = (242)(52) Vouter = 94,096.98 Vinner = (222)(52) Vinner = 79,067.60 V = Vouter – Vinner V  15,029.4 in3 44 52 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

21 Geometry 11.5 Volume of Prisms and Cylinders
Example 6 4 5 L A metal bar has a volume of 2400 cm3. The sides of the base measure 4 cm by 5 cm. Determine the length of the bar. Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

22 Geometry 11.5 Volume of Prisms and Cylinders
Example 6 Solution 4 5 L Method 1 V = Bh B = 4  5 = 20 2400 = 20h h = 120 cm Method 2 V = L  W  H 2400 = L  4  5 2400 = 20L L = 120 cm Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

23 Geometry 11.5 Volume of Prisms and Cylinders
Example 7 Diameter = 7 3 in V = 115 in3 A 3-inch tall can has a volume of 115 cubic inches. Find the diameter of the can. Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

24 Geometry 11.5 Volume of Prisms and Cylinders
Summary The volumes of prisms and cylinders are essentially the same: V = Bh & V = r2h where B is the area of the base, h is the height of the prism or cylinder. Use what you already know about area of polygons and circles for B. Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

25 Geometry 11.5 Volume of Prisms and Cylinders
B h h V = r2h V = Bh Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

26 Geometry 11.5 Volume of Prisms and Cylinders
Which Holds More? This one! 3.2 in 1.6 in 4 in 4.5 in 2.3 in Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

27 Geometry 11.5 Volume of Prisms and Cylinders
What would the height of cylinder 2 have to be to have the same volume as cylinder 1? r = 3 r = 4 #2 #1 8 h Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

28 Geometry 11.5 Volume of Prisms and Cylinders
Solution r = 4 #1 8 Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

29 Geometry 11.5 Volume of Prisms and Cylinders
Solution r = 3 #2 h Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

30 Similar Solids We learned that the ratio of the areas is the square of the ratio of the sides in lesson 8.1. 5 125 ? 8 320 A= 320 Find Area of smaller polygon February 16, 2019 Geometry 8.1 Similar Polygons

31 Geometry 11.5 Volume of Prisms and Cylinders
Similar Solids The ratio of the volume of similar figures is the cube of the ratio of sides. The solids are similar. Find the volume of solid B V = Monday, May 5, 2:51 Geometry 11.5 Volume of Prisms and Cylinders

How do you find the volume of prisms and cylinders?

The volume of the prism is V = Bh = πr2h, so the volume of the cylinder is also V = Bh = πr2h. where B is the area of a base, h is the height, and r is the radius of a base. Density is the amount of matter that an object has in a given unit of volume.

How do you find the volume of prisms?

The formula for the volume of a prism is V=Bh , where B is the base area and h is the height. The base of the prism is a rectangle. The length of the rectangle is 9 cm and the width is 7 cm. The area A of a rectangle with length l and width w is A=lw .

What is the formula for volume of cylinders?

Solution: We know the volume of a cylinder is given by the formula – π r² h, where r is the radius of the cylinder and h is the height. = 3.14 x 50² x 100 = 785,000 cm³.