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Volumes Lesson 7.2.1

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**Volumes 7.2.1 California Standards: What it means for you: Key words:**

Lesson 7.2.1 Volumes California Standards: Measurement and Geometry 2.1 Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. Mathematical Reasoning 2.2 Apply strategies and results from simpler problems to more complex problems. Mathematical Reasoning 3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems. What it means for you: You’ll be reminded what volume is, and then see how you can work out the volume of prisms and cylinders. Key words: volume cubic units prism cylinder

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Lesson 7.2.1 Volumes The volume of a 3-D object, like a box, a swimming pool, or a can, is a measure of the amount of space that’s contained inside it. Volume is measured in units like cubic feet (ft3) or cubic centimeters (cm3). This Lesson, you’ll learn how to find the volume of prisms and cylinders.

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**Volumes 7.2.1 Volume Measures Space Inside a Figure**

Lesson 7.2.1 Volumes Volume Measures Space Inside a Figure The amount of space inside a 3-D figure is called the volume. Volume is measured in cubic units. 1 One cubic unit is the volume of a unit cube — a cube with a side length of 1 unit. 4 unit cubes 8 unit cubes Volume = 8 cubic units The number of unit cubes that could fit inside a solid shape and fill it completely is the volume in cubic units.

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**Volumes 7.2.1 The Volume of a Prism is a Multiple of its Base Area**

Lesson 7.2.1 Volumes The Volume of a Prism is a Multiple of its Base Area You can work out the volume of a prism from the area of its base. Base The base is made of 5 unit squares. So it has an area of 5 square units. Height = 3 units Height = 2 units Height = 1 unit When the prism’s height is 2 units, it has a volume of 10 cubic units because it would take 10 unit cubes to make it. When the prism’s height is 1 unit, it has a volume of 5 cubic units because it would take 5 unit cubes to make it. When the prism’s height is 3 units, it has a volume of 15 cubic units. Every time you increase the height by 1 unit you add an extra 5 unit cubes.

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**Volumes 7.2.1 Guided Practice**

Lesson 7.2.1 Volumes Guided Practice 1. The figure below is constructed from unit cubes. What is its volume? 21 unit3 Solution follows…

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**Volumes 7.2.1 Guided Practice**

Lesson 7.2.1 Volumes Guided Practice A prism is one yard high. It has volume 4 yd3. 2. What is the area of the prism’s base? 3. A prism with an identical base has a volume of 16 yd3. How tall is this prism? 4 yd2 4 yards Solution follows…

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**Volumes 7.2.1 Area Formulas Help Work Out the Volume of a Prism**

Lesson 7.2.1 Volumes Area Formulas Help Work Out the Volume of a Prism When you count the number of unit cubes that make a shape, you find the number of unit cubes that make up the base layer, and multiply it by the height in units. You can’t always count the number of unit cubes that are inside a shape, because not all shapes can fit an exact number of unit cubes inside them. Instead you can work out the volume of any prism by multiplying the area of the base by the height. Volume of prism = Base area × Height

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**Volumes 7.2.1 3 in What is the volume of this prism? 7 in Solution**

Lesson 7.2.1 Volumes Example 1 3 in 7 in 2 in What is the volume of this prism? Solution The base of this prism is a triangle. So use the area of a triangle formula to work out its area. Area of base = bh = × 2 × 3 = 3 in2. 1 2 Then just multiply that area by the height of the prism. Volume of prism = base area × height = 3 × 7 = 21 in3. Solution follows…

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**The volume is the same whichever way the prism stands.**

Lesson 7.2.1 Volumes It doesn’t matter if the prism looks like it is lying down — the same method of finding volume can still be used. The volume is the same whichever way the prism stands. The base is always the shape that is the same through the entire prism — it’s just not always at the bottom.

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**What is the volume of this prism?**

Lesson 7.2.1 Volumes Example 2 5 yd 2 yd 1 yd What is the volume of this prism? Solution Treat the triangle as the base of the prism, and the length of 5 yards as the height. Remember, the base is always the shape that is the same through the entire prism. Area of base = bh = × 1 × 2 = 1 yd2. 1 2 So, volume of prism = base area × height = 1 × 5 = 5 yd3. Solution follows…

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**Volumes 7.2.1 Guided Practice**

Lesson 7.2.1 Volumes Guided Practice Work out the volumes of the figures in Exercises 4–6. 3 m 11 m 1 ft 3 ft 4 in 2 in 9 in ½ × 1 × 1 = 0.5 0.5 × 3 = 1.5 ft3 ½ × 4 × 2 = 4 4 × 9 = 36 in3 3 × 3 × 11 = 99 m3 Solution follows…

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**Volumes 7.2.1 Find the Volume of a Cylinder in the Same Way**

Lesson 7.2.1 Volumes Find the Volume of a Cylinder in the Same Way Circular cylinders are similar to prisms — the only difference is that the base is a circle instead of a polygon. So you can work out the volumes of cylinders in the same way as the volumes of prisms — by multiplying the base area by the height. You use the area of a circle formula to get the base area of a cylinder. Area of a circle = p × radius2

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**Volumes 7.2.1 What is the volume of this cylinder? Use p = 3.14.**

Lesson 7.2.1 Volumes Example 3 What is the volume of this cylinder? Use p = 3.14. 4 cm 15 cm Solution Area of base = pr2 = p × 42 = p × 16 = cm2. Height = 15 cm. So volume of cylinder = × 15 = cm3. Solution follows…

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**Volumes 7.2.1 Guided Practice**

Lesson 7.2.1 Volumes Guided Practice Work out the volumes of the figures in Exercises 7–9. Use p = 3.14. 4 in 10 in 1 yd 8 yd 2 cm 5 cm 3.14 × 42 = 50.24 50.24 × 10 = in3 3.14 × 12 = 3.14 3.14 × 8 = yd3 3.14 × 22 = 12.56 12.56 × 5 = 62.8 cm3 Solution follows…

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**Volumes 7.2.1 Rectangular Prisms and Cubes are Special Cases**

Lesson 7.2.1 Volumes Rectangular Prisms and Cubes are Special Cases The area of the base of a rectangular prism is: length (l) × width (w) If you multiply that by height to get the volume then you get: Volume = length (l) × width (w) × height (h) l w h V (rectangular prism) = lwh

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**Volumes 7.2.1 What is the volume of this rectangular prism? Solution**

Lesson 7.2.1 Volumes Example 4 What is the volume of this rectangular prism? 13 ft 20 ft 6 ft Solution Volume = lwh = 13 × 20 × 6 = 1560 ft3. Solution follows…

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**Volumes 7.2.1 All sides of a cube are the same length.**

Lesson 7.2.1 Volumes All sides of a cube are the same length. s For a cube with side length s, the base area is s × s = s2, and the height is also s, so the volume is s2 × s = s3. V (cube) = s3 where s is the side length.

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**Volumes 7.2.1 What is the volume of this cube? Solution**

Lesson 7.2.1 Volumes Example 5 7 in What is the volume of this cube? Solution Volume = s3 = 73 = 343 in3. Solution follows…

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**Volumes 7.2.1 Guided Practice**

Lesson 7.2.1 Volumes Guided Practice Work out the volumes of the figures in Exercises 10–12. Figures with only one side length shown are cubes. 12. 8 cm 2 cm 8 in 8 × 2 × 2 = 32 cm3 83 = 512 in3 5 yd 8 yd 3 yd 8 × 5 × 3 = 120 yd3 Solution follows…

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**Volumes 7.2.1 Independent Practice**

Lesson 7.2.1 Volumes Independent Practice 1. The figure on the right is constructed from cubes with a volume of 1 in3. What is its volume? 2. How many unit cubes can you fit inside a figure with dimensions 3 units × 3 units × 5 units? 34 in3 45 Solution follows…

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**Volumes 7.2.1 Independent Practice**

Lesson 7.2.1 Volumes Independent Practice 3. What is the volume of the prism shown on the right? 4. A cylinder of volume 32 in3 is cut in half. What is the volume of each half? 35,000 ft3 Base area = 500 ft2 Height = 70 ft 16 in3 Solution follows…

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**Volumes 7.2.1 Independent Practice**

Lesson 7.2.1 Volumes Independent Practice Work out the volumes of the figures shown in Exercises 5–7. Use p = 8. What is the volume of a cube with side length 3 yd? 5 in 2 in 7 in 8 cm 10 cm 50 cm 2 yd 5 yd 70 in3 2000 cm3 62.8 yd3 27 yd3 Solution follows…

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Lesson 7.2.1 Volumes Round Up Volume is the amount of space inside a 3-D figure, and it’s measured in cubed units. For cylinders and prisms, you can multiply the base area by the height of the shape to find the volume.

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