SAT Math · Geometry and Trigonometry

Volume of a cone and a pyramid

Q: What is the formula for the volume of a pyramid?

V = ⅓Bh, where B is the area of the base and h is the perpendicular height. For a rectangular base, B = lw, so V = ⅓lwh.

Q: Is the volume of a cone formula on the SAT reference sheet?

Yes. The SAT reference sheet in Bluebook includes V = ⅓πr²h for a cone and V = ⅓lwh for a rectangular pyramid, along with the volumes of a sphere, a cylinder, and a rectangular prism.

Q: Why is the volume of a cone one third of a cylinder?

A cone is a pyramid with a circular base, and every pyramid holds one third of the prism with the same base and height. You can see it with a cube: three identical square pyramids that meet at one corner fill the cube exactly.

Q: How do you find the height of a cone from its volume?

Rearrange the formula to h = 3V ÷ (πr²). Multiply the volume by 3, then divide by π times the radius squared.

Q: How can Aristotle help with SAT geometry?

Aristotle is a voice AI SAT tutor. Your child talks through a volume question out loud, and Aristotle asks guiding questions instead of giving the answer, starting from the step where their reasoning broke. It tracks 113 SAT Math skills, including area and volume.

Why both formulas have a one third in them, and how to use them on SAT Math.

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Updated October 30, 2026

$V = \tfrac{1}{3}\pi r^2 h \qquad V = \tfrac{1}{3}Bh$

$h$ is the straight-up height. The slant height $\ell$ runs along the side and is longer.

What is the volume of a cone and a pyramid?

V = \tfrac{1}{3}\,B\,h

B = \pi r^2 \text{ for a cone}

Anything that rises from a flat base to a single point holds one third of base area times height. For a cone the base is a circle, $\pi r^2$ . For a pyramid with a rectangular base, $B = lw$ .

Same rule for both: a third of the base area, times the height.

Why one third: the cone lives inside a cylinder

V_{\text{cone}} = \tfrac{1}{3}\,\pi r^2 h = \tfrac{1}{3}\,V_{\text{cylinder}}

Stand the cone inside a cylinder with the same base and height. The cone tapers to a point, so most of the cylinder is empty around it: two thirds, exactly. Fill the cone with water and pour; three cones fill the cylinder.

Same base, same height: the cone is one third of the cylinder.

Height is straight up, not down the side

h^2 + r^2 = \ell^2

h = \sqrt{\ell^2 - r^2}

The height $h$ drops from the tip straight to the center of the base. The slant $\ell$ runs down the outside and is always longer. If a question gives $\ell$ , the radius, height and slant form a right triangle; find $h$ first.

r, h and ℓ form a right triangle inside the cone.

Why there is a one third

Answer each question in your head, then scroll for the answer.

Think it through

Question 1
Pick one corner of a cube and connect it to every other corner. How many pyramids do you get?
Answer
Three identical pyramids, each with a face of the cube as its base and the cube's side as its height.
Question 2
So what fraction of the cube is each pyramid?
Answer
One third. That is $V = \tfrac{1}{3}Bh$ , and it holds for any base shape.
Question 3
A cone is a pyramid with a circular base. What is its volume?
Answer
$\tfrac{1}{3} \times \pi r^2 \times h = \tfrac{1}{3}\pi r^2 h$ . Three full cones fill the matching cylinder.

Try it yourself

Work it out first, then check your answer. SAT-style questions, not official College Board questions.

Question 1

A cone has a radius of 3 centimeters and a height of 8 centimeters. What is its volume, in cubic centimeters?

Question 2

A right rectangular pyramid has a base that is 6 inches by 4 inches. Its volume is 64 cubic inches. What is its height, in inches?

Question 3

A cone has a base radius of 5 and a slant height of 13. What is the volume of the cone?

Question 4

The radius of a cone is doubled and its height stays the same. By what factor does the volume increase?

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The formulas

Cone

$V = \tfrac{1}{3}\pi r^2 h$

$r$ is the base radius. $h$ is the straight-up height.

Pyramid

$V = \tfrac{1}{3}Bh$

$B$ is the base area. For a rectangle, $V = \tfrac{1}{3}lwh$ .

Cylinder, for comparison

$V = \pi r^2 h$

Same base and height, no $\tfrac{1}{3}$ . A cone is one third of it.

Height from slant height

$h = \sqrt{\ell^2 - r^2}$

Height, radius, and slant height form a right triangle.

Try it

Change the radius and the height

Drag the sliders. The cone redraws to scale and its volume updates.

Radius r3

Height h8

ConeV = ⅓π × 3² × 8 = 24π ≈ 75.4

Cylinder, same base and heightV = π × 3² × 8 = 72π, three times the cone

Double the radius and the volume goes up 4 times. Double the height and it only doubles.

Common mistakes

Dropping the one third

Without it you get the cylinder's volume, three times too big.

Using the slant height as the height

Given the slant height, find the real height with the Pythagorean theorem first.

Using the diameter as the radius

A base 10 across has radius 5. Using 10 makes it four times too big.

Forgetting to square the radius

The formula is $\tfrac{1}{3}\pi r^2 h$ , not $\tfrac{1}{3}\pi r h$ . Square the radius before you multiply by anything else.

How it shows up on the SAT

Geometry and Trigonometry is about 15% of SAT Math. Expect a missing radius or height, a slant height you must turn into a height, or a scaling question.

What Aristotle tracks in area and volume

5 SAT Math skills, including:

Finding area and perimeter from given information
Finding surface area and volume from given information
Using scale factors for length, area, and volume
Choosing and using area formulas accurately

What is the formula for the volume of a cone?

V = ⅓πr²h, where r is the radius of the circular base and h is the perpendicular height. It is one third of the volume of a cylinder with the same radius and height.

What is the formula for the volume of a pyramid?

Is the volume of a cone formula on the SAT reference sheet?

Why is the volume of a cone one third of a cylinder?

How do you find the height of a cone from its volume?

How can Aristotle help with SAT geometry?

More SAT concepts

Surface area of a sphere

The one sphere formula the SAT reference sheet leaves out, and how to use it.

The Pythagorean theorem

a² + b² = c² for any right triangle, plus the triples and special triangles that save time on SAT Math.

Equation of a circle

Read the center and radius from (x − h)² + (y − k)² = r², and complete the square when the equation is expanded.

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