Step-by-step examples for comparing volumes of similar 3D shapes Grade 8

📚 Quick Study Guide

• 📐 Similar Shapes: Similar 3D shapes have the same shape but different sizes. Their corresponding dimensions are proportional.
• 📏 Volume Scale Factor: If the length scale factor between two similar shapes is $k$, then the volume scale factor is $k^3$.
• ➗ Calculating Volume: To compare volumes, find the volume of each shape first using the appropriate formula (e.g., $V = lwh$ for a rectangular prism, $V = \pi r^2 h$ for a cylinder).
• 🧪 Ratio of Volumes: Divide the volume of the larger shape by the volume of the smaller shape to find the ratio. This ratio is equal to $k^3$.
• 💡 Finding Unknown Volumes: If you know the volume of one shape and the length scale factor, you can find the volume of the other shape by multiplying or dividing by $k^3$.

Practice Quiz

1. A rectangular prism has dimensions 3cm x 4cm x 5cm. A similar prism has dimensions scaled by a factor of 2. What is the volume of the larger prism?
   A) 60 cm³
   B) 120 cm³
   C) 480 cm³
   D) 960 cm³
2. Two spheres are similar. The radius of the smaller sphere is 3 cm, and the radius of the larger sphere is 6 cm. What is the ratio of their volumes?
   A) 2:1
   B) 4:1
   C) 8:1
   D) 16:1
3. A cylinder has a radius of 2 inches and a height of 5 inches. A similar cylinder has a radius of 4 inches. What is the height of the larger cylinder?
   A) 2.5 inches
   B) 5 inches
   C) 10 inches
   D) 20 inches
4. The volume of a small cube is 8 cm³. A larger, similar cube has sides that are twice as long. What is the volume of the larger cube?
   A) 16 cm³
   B) 24 cm³
   C) 32 cm³
   D) 64 cm³
5. A cone has a radius of 1 meter and a height of 3 meters. A similar cone has a radius of 3 meters. What is the volume of the larger cone, given that the volume of a cone is $V = \frac{1}{3}\pi r^2 h$?
   A) $3\pi$ m³
   B) $9\pi$ m³
   C) $27\pi$ m³
   D) $81\pi$ m³
6. Two similar pyramids have volumes of 27 cm³ and 64 cm³. What is the ratio of their corresponding heights?
   A) 3:4
   B) 9:16
   C) 27:64
   D) 8:27
7. A rectangular prism has a volume of 125 cm³. A similar prism has sides that are 3 times longer. What is the volume of the larger prism?
   A) 375 cm³
   B) 1125 cm³
   C) 3375 cm³
   D) 15625 cm³