Integrated Math 3 / Unit 7: Geometric Modeling
7.2 Volume of Pyramids, Cones and Spheres
Notes
- Pyramid: $\displaystyle V = \frac{1}{3}Bh$.
- Cone: $\displaystyle V = \frac{1}{3}\pi r^2 h$.
- Sphere: $\displaystyle V = \frac{4}{3}\pi r^3$.
- A cone has exactly $\displaystyle \frac{1}{3}$ the volume of a cylinder with the same base and height.
- Example: a cone with $r = 3$ and $h = 12$ has $\displaystyle V = \frac{1}{3}\pi(9)(12) = 36\pi$.
Practice
A square pyramid has a base side of 6 and a height of 5. What is its volume?
$\displaystyle V = \frac{1}{3}(36)(5) = 60$.
What is the volume of a sphere with radius 6?
$\displaystyle V = \frac{4}{3}\pi(216) = 288\pi$.
A cone and a cylinder have the same radius and height. The cylinder's volume is $60\pi$. What is the cone's volume?
The cone is one third of the cylinder: $\displaystyle \frac{60\pi}{3} = 20\pi$.
Ready for more? Take the Unit 7 test.