# Finding the Volume and Surface Area of a Right Circular Cone

Volume and Surface Area of a Right Circular Cone

## Volume and Surface Area of a Right Circular Cone

The volume of a cone is found by $$V = \frac{1}{3}πr^2h$$. The surface area of a cone is found by $$S = πr^2 + πr \sqrt{r^2 + h^2}$$. Let’s look at an example.

Find the volume and surface area of the cone in terms of π and to the nearest whole number. We’ll start with volume. First we’re going to write our volume formula.
$$V = \frac{1}{3} πr^2h$$.

Now we need to substitute our values.
$$V = \frac{1}{3} π (5\text{ in})^2(12\text{ in})$$.

Now we’ll simplify that according to PEMDAS.
$$5\text{ in}^2=25\text{ in}^2$$.

Now that all we have left is multiplication we can multiply in any order we want. We could first take $$\frac{1}{3}12=4$$ or $$12\text{ in} \div 3=4\text{ in}$$.

$$4\text{ in}\times25\text{ in}^2=100\text{ in}^3$$
$$100\text{ in}^3 \times π = 100π\text{ in}^3$$

So there’s our answer in terms of π.

Now for the nearest whole number, we multiply times π and we get that our volume is $$314\text{ inches}^3$$. There are two answers for volume of that cone.

Now to find surface area. We’ll start with our formula surface area:

$$S = πr^2 + πr \sqrt{r^2 + h^2}$$.

Then we need to substitute our radius and our height.
$$S = π(5in)^2 + π(5in) \sqrt{(5 in)^2 + (12 in)^2}$$.

Now we need to simplify.

$$5\text{ in}^2π=25\text{ in}^2 \times \pi=25\pi\text{ in}^2$$
$$(5\text{ in})\pi=5\pi\text{ in}$$
$$25\pi\text{ in}^2+5\pi\text{ in}\sqrt{(5 in)^2 + (12 in)^2}$$

Then we’re going to simplify under our radical.
$$\sqrt{5\text{ in}^2+12\text{ in}^2}=\sqrt{25\text{ in}^2+144\text{ in}^2}$$

We still need to simplify under our radical so the rest of this isn’t going to change.
$$S=25π\text{ in}^2+5π\text{ in}\sqrt{169\text{ in}^2}$$

Then we need to take the square root of that.
$$S=25π\text{ in}^2+5\pi\text{ in}(13\text{ in})$$

Now we can multiply $$5\pi\text{ in}\times 13\text {in}$$.
$$S= 25π\text{ in}^2+65π\text{ in}^2$$

$$S= 25π\text{ in}^2+65π\text{ in}^2=90π\text{ in}^2$$

That’s our answer in terms of π and then to find it to the nearest whole number we need to multiply times π.
$$90 \times π = 283\text{in}^2$$.

## Practice Questions

Question #1:

What is the volume to the nearest whole number of a cone that has a height of 14 inches and a radius of 6 inches?

535 in3

582 in3

528 in3

498 in3

The correct answer is 528 in3. Start with the volume formula for a cone, which is $$V=\frac{1}{3}πr^2h$$, and plug in 6 for r and 14 for h.
$$V=\frac{1}{3}π(6)^2(14)$$
$$V=\frac{1}{3}π(36)(14)$$
$$V=\frac{1}{3}π(504)$$
$$V=168π$$
$$V=168(3.14159)=527.8$$
$$V=528\text{ in}^3$$

Question #2:

What is the volume in terms of pi of a cone that has a height of 12 cm and a radius of 9 cm?

324π cm3

346π cm3

445π cm3

389π cm3

The correct answer is 324π cm3. Start with the volume formula for a cone and plug in 9 for r and 12 for h.

$$V=\frac{1}{3}π(9)^2(12)$$
$$V=\frac{1}{3}π(81)(12)$$
$$V=\frac{1}{3}π(972)$$
$$V=324π\text{ cm}^3$$

Question #3:

Find the volume of a cone that has a radius of 6 meters and a height of 11 meters. Express your answer in terms of pi.

227π m3

432π m3

145π m3

132π m3

The correct answer is 132π m3. Start with the volume formula for a cone and plug in 6 for r and 11 for h.

$$V=\frac{1}{3}π(6)^2(11)$$
$$V=\frac{1}{3}π(36)(11)$$
$$V=\frac{1}{3}π(396)$$
$$V=132π\text{ m}^3$$

Question #4:

Find the volume of a cone that has a diameter of 18 meters and a height of 30 meters. Express your answer to the nearest whole number.

2,676 m3

2,304 m3

2,499 m3

2,545 m3

The correct answer is 2,545 m3. If the diameter is 18 meters, then the diameter is 9 meters. Start with the volume formula for a cylinder and plug in 9 for r and 30 for h.
$$V=\frac{1}{3}π(9)^2(30)$$
$$V=\frac{1}{3}π(81)(30)$$
$$V=\frac{1}{3}π(2,430)$$
$$V=810π$$
$$V=810(3.14159)=2,544.69$$
$$V=2,545\text{ m}^3$$

Question #5:

Find the volume of a cone that has a radius of 6 yards and a height of 10 yards. Express your answer to the nearest whole number.

488 yd3

377 yd3

366 yd3

411 yd3

The correct answer is 377 yd3. Start with the volume formula for a cylinder and plug in 6 for r and 10 for h.
$$V=\frac{1}{3}π(6)^2(10)$$
$$V=\frac{1}{3}π(36)(10)$$
$$V=\frac{1}{3}π(360)$$
$$V=120π$$
$$V=120(3.14159)=376.99$$
$$V=377\text{ yd}^3$$

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by Mometrix Test Preparation | Last Updated: June 2, 2021