Lesson 5: Volumes of Similar Solids

👨🏽‍🏫 Introduction

Imagine you have two water bottles — one small and one large — with the exact same shape. If the larger one is twice as tall, does it hold twice as much water? 🤔 Actually, no! When solids are similar, their volumes don’t just double when their lengths double — they grow much faster!

This lesson explores how the volumes of similar solids relate to their linear scale factors. You’ll learn how to compare volumes using ratios and solve real-life problems in scaling objects and models.

• 🔁 Understand what makes solids “similar”
• 📏 Learn how volume changes with scale
• ⚙️ Use formulas to find unknown volumes of scaled solids

🧠 Key Concepts

📌 1. What are Similar Solids?

Solids are similar if they have the same shape but different sizes — like model cars or resized water tanks. Their corresponding lengths, widths, and heights are in the same ratio (called the linear scale factor).

📐 2. Volume and Scale Factor

If the linear scale factor between two similar solids is , then:

• Ratio of areas =
• Ratio of volumes =

🧪 3. Formula

If two solids are similar, then:

or solve directly using:

🔍 Worked Examples

1. 1. Example 1: Two similar cones have heights in the ratio 2:5. If the smaller cone has volume 48 cm³, find the volume of the larger cone.

      Step 1: Scale factor = .
      Step 2: Volume ratio = .
      Step 3:
      🟢 The larger cone has a volume of 750 cm³.

1. 1. Example 2: Two similar cylinders have volumes in the ratio 8:27. What is the ratio of their heights?

      Step 1: Volume ratio = .
      Step 2: Take cube root of both sides: .
      🟢 The ratio of their heights is 2:3.

1. Example 3: A small sphere has radius 4 cm and volume 268 cm³. What is the volume of a similar sphere with radius 6 cm?

   Step 1: Scale factor = .
   Step 2: Volume ratio = .
   Step 3:
   🟢 Volume of the larger sphere ≈ 902.25 cm³.

✍🏽 Practice Exercises

🅰️ Section A – Finding Volume

1. Two similar cubes have side lengths in the ratio 1:3. If the volume of the smaller cube is 20 cm³, find the volume of the larger cube.
2. A toy cylinder is scaled up by a factor of 4. If the small one has a volume of 25 cm³, what is the volume of the large one?

🅱️ Section B – Finding Scale Factor

1. Two similar cones have volumes of 250 cm³ and 1000 cm³. What is the ratio of their heights?

🅾️ Section C – Application

1. A smaller tank has radius 3 m and volume 254 m³. A larger tank of the same shape has radius 6 m. Find its volume.

✅ Answers and Explanations

• A1: Scale = 3 → Volume ratio = 3³ = 27 →
• A2: Volume ratio = 4³ = 64 →
• B1: → scale = → height ratio ≈ 1:1.59
• C1: Scale = → Volume ratio = 2³ = 8 →

📌 Recap

• ✅ Similar solids have equal shape but different sizes
• ✅ Volume changes by the cube of the scale factor
• ✅ Use to compare or find unknown volumes

💭 Reflection Prompt

Have you ever used a smaller model to design something bigger — like a building, bridge, or sculpture? How would you use what you’ve learned to estimate the volume or material needed for the full-sized version?