Pre-Algebra
Geometry
Volume of cylinders, cones and spheres
20 practice questions
0 video lessons
Theory + worked examples
Volume of Cylinders, Cones and Spheres
Texas Pre-Algebra (TEKS) • Standard 8.7(A) • Geometry
Volume of Cylinders, Cones and Spheres is a topic in Geometry in the Texas Essential Knowledge and Skills. It is aligned to Standard 8.7(A), which requires students to find the volume of cylinders, cones, and spheres.
Round solids have volume formulas built from \(\pi\): a cylinder is \(\pi r^2h\), a cone one third of that, and a sphere \(\dfrac{4}{3}\pi r^3\).
Theory
Round solids use \(\pi\) in their volume formulas:
- Cylinder: \(V=\pi r^2h\).
- Cone: \(V=\dfrac{1}{3}\pi r^2h\).
- Sphere: \(V=\dfrac{4}{3}\pi r^3\).
A cone is one third of the matching cylinder.
Cylinder, cone, and sphere.
Volume formulas.
Round-solid volumes:
\[V_{\text{cyl}}=\pi r^2h,\ V_{\text{cone}}=\dfrac{1}{3}\pi r^2h,\ V_{\text{sph}}=\dfrac{4}{3}\pi r^3\]
Volume is in cubic units.
How to find round-solid volume
- Identify the solid and its radius.
- Apply the matching formula.
- Substitute \(\pi\approx3.14\) if needed.
- Label in cubic units.
Example 1 — Cylinder
Find the volume of a cylinder with \(r=3,\ h=5\). (\(\pi\approx3.14\))
Solution
\(V=\pi r^2h\).
| \(3.14\cdot9\cdot5\) | \(=\) | \(141.3\) |
Example 2 — Cone
Find the cone's volume with \(r=3,\ h=5\).
Solution
One third of the cylinder.
| \(\dfrac{1}{3}(141.3)\) | \(=\) | \(47.1\) |
Example 3 — Sphere
Find the sphere's volume with \(r=3\).
Solution
\(V=\dfrac{4}{3}\pi r^3\).
| \(\dfrac{4}{3}(3.14)(27)\) | \(=\) | \(113.0\) |
Example 4 — Compare
How does a cone compare to a cylinder of the same base and height?
Solution
The cone is one third the volume.
Common pitfalls
A cone has the \(\dfrac13\); a cylinder does not.
A sphere uses \(r^3\), not \(r^2\).
Volume is cubed units.
Frequently asked questions
What is a cylinder's volume?
\(\pi r^2h\).
What is a cone's volume?
One third \(\pi r^2h\).
What is a sphere's volume?
\(\dfrac{4}{3}\pi r^3\).
How does a cone compare to a cylinder?
One third the volume.
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