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 \(5\pi\text{ in}\times 13\text {in}\).
\(S= 25π\text{ in}^2+65π\text{ in}^2\)
 
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 in3

582 in3

528 in3

498 in3

Answer:

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

Answer:

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

Answer:

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

Answer:

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

Answer:

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