An Ice Cream Cone

A waffle cone has a radius of 3 cm and a height of 10 cm. Find its volume in terms of $\pi$, then with $\pi \approx 3.14$, rounded to the nearest tenth.

Write the formula: $V = \frac{1}{3}\pi r^2 h$ = Formula ready

Square the radius: $r^2 = 3^2 = 9$ = $r^2 = 9$

Multiply by the height and divide by 3: $\frac{1}{3} \times 9 \times 10 = \frac{90}{3} = 30$ = $V = 30\pi$ cm³

Use $\pi \approx 3.14$: $30 \times 3.14 = 94.2$ = $V \approx 94.2$ cm³

Answer: The cone holds exactly $30\pi$ cm³, which is about $94.2$ cm³ of ice cream (not counting the scoop on top!).

Cone Versus Cylinder

A cylinder and a cone both have a diameter of 8 cm and a height of 9 cm. Find both volumes in terms of $\pi$ and compare them.

Get the radius from the diameter: $r = 8 \div 2 = 4$ cm = $r = 4$ cm

Volume of the cylinder: $V = \pi r^2 h = \pi \times 16 \times 9 = 144\pi$ = $144\pi$ cm³

Volume of the cone (one third): $V = \frac{1}{3} \times 144\pi = 48\pi$ = $48\pi$ cm³

Compare: $144\pi \div 48\pi = 3$ = The cylinder holds 3 times as much

Answer: The cylinder holds $144\pi$ cm³ and the cone holds $48\pi \approx 150.72$ cm³. Three cones fill one cylinder.

A Pile of Sand

A pile of sand forms a cone with a diameter of 6 m and a height of 2 m. How many cubic meters of sand are in the pile? Use $\pi \approx 3.14$ and round to the nearest hundredth.

Find the radius: $r = 6 \div 2 = 3$ m = $r = 3$ m

Substitute into the formula: $V = \frac{1}{3} \times 3.14 \times 3^2 \times 2$ = Values plugged in

Work out $\frac{1}{3} r^2 h$ first: $\frac{9 \times 2}{3} = \frac{18}{3} = 6$ = $V = 6\pi$

Multiply by 3.14: $6 \times 3.14 = 18.84$ = $V \approx 18.84$ m³

Answer: The pile contains about $18.84$ m³ of sand (exactly $6\pi$ m³).

Using the Slant Height

A cone has a radius of 6 cm and a slant height of 10 cm. Find its volume in terms of $\pi$ and with $\pi \approx 3.14$, rounded to the nearest hundredth.

Notice that 10 cm is the slant height, not the height: Radius, height and slant height form a right triangle: $l^2 = r^2 + h^2$ = We need $h$ first

Use the Pythagorean theorem: $h^2 = 10^2 - 6^2 = 100 - 36 = 64$ = $h = \sqrt{64} = 8$ cm

Apply the volume formula: $V = \frac{1}{3}\pi \times 6^2 \times 8 = \frac{36 \times 8}{3}\pi = 96\pi$ = $V = 96\pi$ cm³

Use $\pi \approx 3.14$: $96 \times 3.14 = 301.44$ = $V \approx 301.44$ cm³

Answer: $V = 96\pi \approx 301.44$ cm³. Using 10 cm as the height would have given a wrong answer of $120\pi$ cm³.

Finding the Height of a Funnel

A cone-shaped funnel holds 314 cm³ when full. Its radius is 5 cm. How tall is it? Use $\pi \approx 3.14$.

Write the formula with what you know: $314 = \frac{1}{3} \times 3.14 \times 5^2 \times h$ = Only $h$ is unknown

Multiply both sides by 3: $942 = 3.14 \times 25 \times h$ = $942 = 78.5h$

Divide both sides by 78.5: $h = 942 \div 78.5 = 12$ = $h = 12$ cm

Check: $\frac{1}{3} \times 3.14 \times 25 \times 12 = 3.14 \times 100 = 314$ = Correct

Answer: The funnel is $12$ cm tall. In general, $h = \frac{3V}{\pi r^2}$.

Mistake: Forgetting the $\frac{1}{3}$ and using $V = \pi r^2 h$

Why: That is the cylinder formula. A cone with the same base and height holds only one third of the cylinder, so the answer comes out 3 times too big.

Correct: Always write $V = \frac{1}{3}\pi r^2 h$ and divide by 3. For $r = 3$, $h = 10$: $\frac{1}{3}\pi \times 9 \times 10 = 30\pi$, not $90\pi$.

Mistake: Using the diameter instead of the radius

Why: The radius is half the diameter. Squaring the diameter makes the answer 4 times too big.

Correct: If the problem gives the diameter, halve it first: a diameter of 8 cm means $r = 4$ cm and $r^2 = 16$, not $64$.

Mistake: Using the slant height as the height

Why: The slant height runs along the sloping side, so it is always longer than the perpendicular height. Using it makes the volume too big.

Correct: The height goes straight from the apex down to the center of the base. If you only know the slant height $l$, find $h$ with $h^2 = l^2 - r^2$.

Mistake: Writing the answer in square units, like cm²

Why: Volume is three-dimensional: $r \times r \times h$ multiplies three lengths.

Correct: Volume is always in cubic units: cm³, m³, in³. Remember that $1$ cm³ $= 1$ mL.

Ice Cream Shops

Ice cream shops choose cone sizes and prices by how much ice cream fits inside each cone.

A small cone with radius 2 cm and height 9 cm holds $\frac{1}{3}\pi \times 4 \times 9 = 12\pi \approx 37.68$ cm³.

Paper Cups at a Water Dispenser

Many office water dispensers use cone-shaped paper cups. They are cheap to make and easy to stack.

A paper cone cup with radius 3 cm and height 10 cm holds $30\pi \approx 94.2$ cm³, which is about $94$ mL of water.

Piles of Sand and Road Salt

When sand, gravel or road salt is poured from above, it forms a cone. Builders and road crews use the cone formula to estimate how much material is in a pile.

A sand pile with radius 3 m and height 2 m contains $6\pi \approx 18.84$ m³ of sand.

Q: Why is it exactly one third, and not one half?

A: A cone gets narrower all the way up, so most of its volume sits near the base. You can test it with water or rice: three full cones fill a cylinder with the same base and height. Later, in calculus, you can prove that the factor is exactly $\frac{1}{3}$. Pyramids follow the same one third rule.

Q: What if the cone is tilted (an oblique cone)?

A: The formula still works, as long as $h$ is the perpendicular height from the apex to the plane of the base. A tilted cone with the same base and the same perpendicular height has the same volume as a straight one.

Q: When should I leave the answer in terms of $\pi$?

A: When the problem asks for an exact answer or says "in terms of $\pi$", write it like $48\pi$ cm³. When you need a number for a real situation, multiply by $3.14$ (or use the $\pi$ key) and round to the place the problem asks for.

Q: How do I find the height if I only know the slant height?

A: The radius, the height and the slant height make a right triangle with the slant height as the hypotenuse. So $h = \sqrt{l^2 - r^2}$. For $r = 5$ and $l = 13$: $h = \sqrt{169 - 25} = \sqrt{144} = 12$.