Solve problems involving volume of solids (e.g., prisms, pyramids, cylinders). | Area and volume | SAT Math

Volume comes from base area times height, with cones and pyramids taking one-third of the matching prism or cylinder.

Core Idea

Volume = (area of the base) × height for prisms and cylinders. Cones and pyramids are exactly $13$ of their prism/cylinder counterpart. Sphere volume is $43 𝜋 𝑟 3$ .

Understanding

Rule: Volume starts from the base area and extends through the height.

Prisms and cylinders use base area $\times$ height.
Cones and pyramids use one-third of the matching prism/cylinder volume.
Spheres use $43 𝜋 𝑟 3$ .

Set up the solid first, then choose the formula that matches it.

Step by Step

Identify the solid and find the base shape.
Compute the area of the base.
Multiply by the height. If the solid is a cone or pyramid, multiply by $13$ .
For a sphere, use $43 𝜋 𝑟 3$ directly.

Misconceptions

Forgetting the $13$ factor for cones and pyramids.
Using diameter instead of radius in $𝜋 𝑟 2 ℎ$ .
Squaring the radius instead of cubing it for sphere volume.

Question

Worked Example

A cone-shaped paper cup has a radius of 3 cm and a height of 8 cm. What is the volume of the cup, in cubic centimeters?

Select an answer to see the explanation