The SAT provides most geometry formulas at the beginning of each math section. But knowing them cold — and knowing when to use each — saves precious time.
Rectangular Prism (Box)
Example 1: A box is 5 cm × 3 cm × 4 cm.
Cylinder
Example 2: A cylinder has radius 3 and height 10.
Cone
Note: a cone is exactly of a cylinder with the same base and height.
Example 3: A cone has radius 4 and height 9.
Sphere
Example 4: A basketball has a radius of approximately 4.7 inches. What is its volume?
Pyramid
where is the area of the base.
Example 5: A square pyramid has a base with side length 6 and height 8.
SAT Application: Comparing Volumes
Example 6: Cylinder A has radius 4 and height 6. Cylinder B has radius 6 and height 4. Which has the greater volume?
Cylinder A:
Cylinder B:
Cylinder B is larger. Doubling the radius has a bigger impact than doubling the height because radius is squared.
Composite Shapes
Example 7: An ice cream cone is a hemisphere (half-sphere) on top of a cone. The radius is 3 cm and the cone height is 10 cm. Find the total volume.
Cone:
Hemisphere:
Total:
Practice Problems
Problem 1: A cylinder has volume . If the radius is 5, what is the height?
Solution
Problem 2: A sphere has surface area . What is the radius?
Solution
Problem 3: A cone and a cylinder share the same base (radius 5) and height (12). How do their volumes compare?
Solution
Cylinder:
Cone:
The cone is the volume of the cylinder.
Key Takeaways
- Cone = cylinder; Pyramid = prism (same base and height)
- Radius is squared in cylinder/cone volumes — changes to radius have a bigger impact than height
- For composite shapes, calculate each part separately and add
- The SAT reference sheet has these formulas, but knowing them saves time
- Always check if the question asks for volume OR surface area
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