# Finding the 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 \(5\pi\text{ in}\times 13\text {in}\).

\(S= 25π\text{ in}^2+65π\text{ in}^2\)

Now finally we can add.

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

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 in^{3}

582 in^{3}

528 in^{3}

498 in^{3}

**Answer:**

The correct answer is 528 in^{3}. 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 cm^{3}

346π cm^{3}

445π cm^{3}

389π cm^{3}

**Answer:**

The correct answer is 324π cm^{3}. 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 m^{3}

432π m^{3}

145π m^{3}

132π m^{3}

**Answer:**

The correct answer is 132π m^{3}. 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 m^{3}

2,304 m^{3}

2,499 m^{3}

2,545 m^{3}

**Answer:**

The correct answer is 2,545 m^{3}. 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 yd^{3}

377 yd^{3}

366 yd^{3}

411 yd^{3}

**Answer:**

The correct answer is 377 yd^{3}. Start with the volume formula for a cylinder and plug in 6 for *r* and 10 for *h*.

\(V=\frac{1}{3}π(6)^2(10)\)

\(V=\frac{1}{3}π(36)(10)\)

\(V=\frac{1}{3}π(360)\)

\(V=120π\)

\(V=120(3.14159)=376.99\)

\(V=377\text{ yd}^3\)