Volume and Cross Sections Notes and Worksheets - Lindsay Bowden - Free Printable

This image is a Volume Graphic Organizer designed to help students understand and calculate the volume of various 3D shapes. The organizer includes five common 3D shapes: rectangular prism (or cube), square pyramid (or rectangular pyramid), cylinder, cone, and sphere. For each shape, it provides:

- A picture with labeled dimensions
- The volume formula
- Notes explaining key components of the formula

Let’s go through each shape, explain the formulas, and fill in the missing blanks in the "NOTES" column where necessary.

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1. Rectangular Prism or Cube

- Picture: A box with dimensions 8 ft (length), 6 ft (width), 7 ft (height)
- Formula: $ V = Bh $
- Notes:
- *B stands for the area of the base*
- *For a rectangular base, change B to ___* → $ l \times w $ (length × width)
- *For a square base, change B to ___* → $ s^2 $ (side squared)
- $ h = $ height

✔ Explanation:
The base area $ B $ depends on the shape of the base.
- If the base is rectangular, then $ B = l \times w $.
- If the base is square, then $ B = s^2 $.
So, for this prism:
$ B = 8 \times 6 = 48 $ ft²
$ h = 7 $ ft
$ V = 48 \times 7 = 336 $ ft³

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2. Square Pyramid or Rectangular Pyramid

- Picture: A pyramid with a square base (5 cm × 5 cm), height = 6 cm
- Formula: $ V = \frac{1}{3}Bh $
- Notes:
- *B stands for the area of the base*
- *For a rectangular base, change B to ___* → $ l \times w $
- *For a square base, change B to ___* → $ s^2 $
- $ h = $ height
- *Height must be perpendicular to base*

✔ Explanation:
Base is square → $ B = 5 \times 5 = 25 $ cm²
$ h = 6 $ cm
$ V = \frac{1}{3} \times 25 \times 6 = 50 $ cm³

> ✔ The height must be measured from the apex straight down to the center of the base (perpendicular).

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3. Cylinder

- Picture: Cylinder with radius = 4 in, height = 10 in
- Formula: $ V = \pi r^2 h $
- Notes:
- $ r = $ radius
- $ h = $ height

✔ Explanation:
Plug in values:
$ V = \pi \times (4)^2 \times 10 = \pi \times 16 \times 10 = 160\pi $ in³
Approximately $ 160 \times 3.14 = 502.4 $ in³

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4. Cone

- Picture: Cone with radius = 2 mm, height = 5 mm
- Formula: $ V = \frac{1}{3}\pi r^2 h $
- Notes:
- $ r = $ radius
- $ h = $ height
- *Height must be perpendicular to base*

✔ Explanation:
$ V = \frac{1}{3} \pi (2)^2 (5) = \frac{1}{3} \pi \times 4 \times 5 = \frac{20}{3}\pi $ mm³ ≈ $ 20.94 $ mm³

> ⚠️ The height must be vertical (from tip to center of base), not slant height.

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5. Sphere

- Picture: Sphere with radius = 6 cm
- Formula: $ V = \frac{4}{3}\pi r^3 $
- Notes:
- $ r = $ radius
- *no height*
- *make sure the radius is CUBED*

✔ Explanation:
$ V = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi \times 216 = 288\pi $ cm³ ≈ $ 904.32 $ cm³

> 🔍 Important: The radius must be cubed, not squared.

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✔ Final Answers for Missing Blanks:

| Shape | Change B to... |
|------|----------------|
| Rectangular Prism | $ l \times w $ |
| Square Base | $ s^2 $ |
| Square Pyramid | $ l \times w $ |
| Square Base | $ s^2 $ |

These are filled in based on the type of base.

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Summary Table of Volume Formulas:

| Shape | Formula | Key Notes |
|-------|--------|---------|
| Rectangular Prism/Cube | $ V = Bh $ | $ B = l \times w $ or $ s^2 $, $ h = $ height |
| Square/Rectangular Pyramid | $ V = \frac{1}{3}Bh $ | Height must be perpendicular; $ B = l \times w $ or $ s^2 $ |
| Cylinder | $ V = \pi r^2 h $ | $ r = $ radius, $ h = $ height |
| Cone | $ V = \frac{1}{3}\pi r^2 h $ | Height must be perpendicular |
| Sphere | $ V = \frac{4}{3}\pi r^3 $ | Radius must be cubed, no height |

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🎯 Final Answer: Fill in the blanks in the NOTES section:

- For rectangular base, change B to $ l \times w $
- For square base, change B to $ s^2 $
→ These apply to both prism and pyramid rows.

✔ This graphic organizer helps students remember formulas and how to adapt them based on the base shape.