Volume of Pyramids, Cones and Spheres
[PATHS extension] Discover the one-third rule, calculate volumes of pyramids, cones and spheres, and solve advanced composite solid problems.
Printable Worksheets
Print or save as PDF — or build a custom worksheet from any module's questions.
Worksheet
Download or print the worksheet to work through this lesson.
Q1: A prism and a pyramid have the same base area and the same height. Before reading the formula, predict how many pyramids of sand would fit inside the prism. Explain your reasoning.
Q2: A sphere of radius 6 cm and a cube of side 6 cm — which do you think has the larger volume? Make a prediction and explain why.
Learning Intentions
Know
- The volume formulas for pyramids, cones and spheres.
- The one-third relationship between prisms and pyramids with the same base and height.
Understand
- Why a pyramid has exactly one-third the volume of a prism with the same base and height.
- How the sphere formula connects to its surface area.
Can Do
- Calculate the volume of pyramids, cones and spheres.
- Solve composite solid problems involving pyramids, cones and spheres.
Success Criteria
- I can calculate the volume of a pyramid, cone and sphere given their dimensions.
- I can use the one-third rule to relate prism and pyramid volumes.
- I can solve composite problems involving these shapes in real-world contexts.
Key Terms
Common Mistakes to Avoid
Wrong: Using the slant height instead of the perpendicular height in the volume formula for a pyramid or cone.
Right: Volume always uses the perpendicular height ($h$), measured from the base to the apex at a right angle.
Wrong: Forgetting the $\frac{1}{3}$ factor for pyramids and cones. $V = \frac{1}{3} \times A_{base} \times h$.
Right: A pyramid with the same base and height as a prism has exactly one-third the prism's volume.
A pyramid or cone with the same base and height as a prism or cylinder has exactly one-third the volume. This is a fundamental geometric relationship.
The $\frac{1}{3}$ factor arises because the cross-sectional area shrinks from the base to zero at the apex. This can be proven using calculus in later years.
What to write in your book
- Pyramid: $V = \frac{1}{3} \times A_{base} \times h$
- Cone: $V = \frac{1}{3} \pi r^2 h$
- Always use perpendicular height, not slant height
- A pyramid has exactly one-third the volume of a prism with the same base and height
Correct! $V = \frac{1}{3}\pi(3)^2(8) = \frac{1}{3}\pi(9)(8) = 24\pi$ cm$^3$.
Not quite. $V = \frac{1}{3}\pi(3)^2(8) = \frac{1}{3}\pi(9)(8) = 24\pi$ cm$^3$.
A sphere is the most efficient shape in nature — it has the smallest surface area for a given volume.
Notice the connection to surface area: $SA = 4\pi r^2$ and $V = \frac{4}{3}\pi r^3$. If you know one, you can find the other.
What to write in your book
- Sphere: $V = \frac{4}{3}\pi r^3$
- Connection to surface area: $SA = 4\pi r^2$ and $V = \frac{4}{3}\pi r^3$
- If you know one, you can find the other
Correct! $V = \frac{4}{3}\pi(6)^3 = \frac{4}{3}\pi(216) = 288\pi$ cm$^3$.
Not quite. $V = \frac{4}{3}\pi(6)^3 = \frac{4}{3}\pi(216) = 288\pi$ cm$^3$.
Interactive: One-Third Relationship Explorer
Your Turn
Question 1: Find the volume of a cone with radius 4 cm and perpendicular height 9 cm.
Question 2: Find the volume of a sphere with diameter 10 cm.
Question 3: A pyramid has a rectangular base 5 cm by 4 cm and height 6 cm. Find its volume.
Revisit Your Thinking
Look back at your Think First prediction about the prism and pyramid. The prism has volume 240 cm$^3$. What is the exact volume of the pyramid with the same base and height? How many pyramids of sand would fit inside the prism? Explain why this makes sense geometrically.
Earlier you were asked: What was your first thought on this topic?
Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?
What is the volume of a cone with radius 3 cm and perpendicular height 8 cm?
A sphere has radius 6 cm. What is its volume?
A square-based pyramid has base side 8 cm and height 6 cm. What is its volume?
A cone and a cylinder have the same radius and the same height. The cylinder has volume $60\pi$ cm$^3$. What is the volume of the cone?
A solid metal ball bearing has radius 0.5 cm. What is its volume?
A conical paper cup has radius 4 cm and perpendicular height 12 cm.
(a) Find the volume of the cup. (2 marks)
(b) Convert this volume to millilitres. (1 mark)
A solid is formed by placing a hemisphere of radius 6 cm on top of a cone of radius 6 cm and perpendicular height 8 cm.
(a) Find the volume of the cone. (2 marks)
(b) Find the volume of the hemisphere. (1 mark)
(c) Find the total volume of the solid. (1 mark)
A rectangular-based pyramid has a base measuring 10 m by 8 m. Its apex is directly above the centre of the base at a height of 6 m.
(a) Find the volume of the pyramid. (2 marks)
(b) A rectangular prism with the same base and height would hold how many times more volume? (1 mark)
(c) Explain why this relationship makes sense. (2 marks)
Pyramid
$V = \frac{1}{3}A_{base}h$
Cone
$V = \frac{1}{3}\pi r^2 h$
Sphere
$V = \frac{4}{3}\pi r^3$
One-third
Pyramid = $\frac{1}{3}$ prism
Perp. height
Not slant height
1 cm$^3$
= 1 mL
Real-Life Link
The one-third rule appears throughout engineering and nature. Egyptian pyramids demonstrate this geometry at monumental scale. Modern architects use pyramidal roofs because they shed water efficiently and require less material than a full prism. Ice cream cones are cones (naturally), and manufacturers calculate cone volumes to standardise serving sizes. Ball bearings are spheres — their volume determines their mass and therefore their inertia in mechanical systems. In Australia, spherical LNG storage tanks at ports like Gladstone use the sphere's optimal volume-to-surface-area ratio to minimise heat transfer and evaporation.
Game Time!
Test your advanced volume skills in an interactive challenge.
Play Advanced Volume Challenge