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

## Volume and Surface Area of a Right Circular Cylinder

Volume is the space that a figure occupies. To find the volume of a right circular cylinder, multiply the area of the base, which is a circle PI r squared, times the height of the cylinder. To find the surface area of a cylinder, which is the area of the surfaces of the cylinder, find the lateral area or the area around the side of the cylinder by multiplying two times pi times the radius times the height. Then add the areas of the bases, which are circles. Two times PI R squared because there are two bases, two circles. Let’s look at this example.

Find the volume and surface area of the following right circular cylinder in terms of pi and to the nearest whole number. We’ll start with volume. First, we’re going to write our formula down. The volume of a cylinder is found by multiplying pi times the radius squared times the height of the cylinder. Then we need to substitute our values. Volume is pi times the radius is the distance from the center to the outside of your circular base. Three feet squared times the height of the cylinder eight feet.

We need to simplify using order of operations so we’re going to square our three feet firs. Volume is three feet squared is nine. Nine feet squared times eight feet is 72 pi feet cubed. The feet cubed came from squaring feet squared here and then times feet. Feet squared times feet, feet cubed. That gives us the answer the first way which is in terms of pi meaning we don’t multiply times pi. Pi must stay in our answer. Then to find it to the nearest whole number, we do multiply times pi. 72 times pi is 226 feet cubed. These would be our two answers for volume of our cylinder.

Now for surface area. Surface area is two times pi times the radius times the height plus two times pi times radius squared. Then we substitute our values. Surface area is two times pi times the radius, three feet, times the height, eight feet, plus two times pi times the radius three feet squared. Now we need to simplify. The surface area is three feet times eight feet is 24 feet squared, times 2 is 48 feet squared, times pi, 48 pi feet squared, plus starting with three feet squared. Three feet squared is nine feet squared times two. 18 feet squared times pi 18 pi feet squared.

Then we can combine our like terms. 48 pi feet squared plus 18 pi feet squared. That gives us 66 pi feet squared. There’s our answer in terms of pi. Then to find it to the nearest whole number, we need to multiply times pi. 66 times pi is 207 feet squared. There’s our surface area in terms of pi and to the nearest whole number

## Frequently Asked Questions

#### Q

### How do you calculate the surface area of a cylinder?

#### A

Calculate the surface area of a cylinder by using the formula \(SA=2πr^2+2πrh\), where *r* is the radius of the circular base and *h* is the height of the cylinder.

Ex. Find the surface area of this cylinder.

Find the surface area by using the formula \(SA=2πr^2+2πrh\). The radius (*r*) measures 2 ft and the height (*h*) measures 6 ft.

\(SA=2π(2\text{ ft})^2+2π(2\text{ ft})(6\text{ ft})\)

\(=8π\text{ ft}^2+24π\text{ ft}^2\)

\(=32π\text{ ft}^2\)

\(≈100.53\text{ ft}^2\)

#### Q

### What is the surface area formula?

#### A

The general formula for surface area of a prism is \(SA=2B+ph\), where *B* is the area of the base, *p* is the perimeter of the base, and *h* is the height of the prism.

Remember, a cylinder is a special kind of prism with circular bases, so you can substitute \(πr^2\) for *B* (area of the base) and \(2πr\) for *p* (perimeter of the base). The formula for surface area of a cylinder is \(SA=2πr^2+2πrh\).

#### Q

### What is the shape of the base of a cylinder?

#### A

The shape of the base of a cylinder is a circle.

## Practice Questions

**Question #1:**

Calculate the surface area of the following cylinder.

41.81 yd^{2}

51.81 yd^{2}

61.81 yd^{2}

71.81 yd^{2}

**Answer:**

The volume of a cylinder can be calculated using the formula \(SA=2πr^2+2πrh\). A cylinder consists of two circles and one rectangle. The surface area of a circle is found using the formula \(πr^2\),so \(2πr^2\) will allow us to find the surface area of the top and bottom faces of the cylinder. The rectangle that wraps around the cylinder has a height that is 10 cm, and a width that is \(2πr\). Essentially, the width of the rectangle is the circumference of the circle.

When the height (*h*), radius (*r*), and an approximation of \(π\) (3.14) are plugged into the formula \(SA=2πr^2+2πrh\), it becomes \(SA=2(3.14)(1.5)^2+2(3.14)(1.5)(4)\). This simplifies to \(SA=14.13+37.68\), which reduces to 51.81. The surface area of the cylinder is 51.81 yd^{2}.

**Question #2:**

Calculate the volume of the following cylinder.

202.9 cm^{3}

302.3 cm^{3}

402.6 cm^{3}

502.4 cm^{3}

**Answer:**

The volume of a cylinder can be calculated using the formula \(V=πr^2(h)\). When determining the volume of a cylinder, you are simply finding the area of the circular base shape and then multiplying this by the height. The radius of the circular base shape in this cylinder is 4 cm, and the height is 10 cm. These two values can be plugged into the formula so that \(V=πr^2(h)\) becomes \(V=π(4)^2(10)\). When the approximation of pi (3.14) is substituted in for the symbol \(π\), the equation simplifies to 502.4. The volume of the cylinder is 502.4 cm^{3}.

**Question #3:**

A grain silo consists of a cylinder with a dome on top. Farmer Jenkis needs to calculate the volume of grain contained in a silo that has a cylinder height of 50 feet and a cylinder diameter of 10 feet. The dome will remain empty. If the cylindrical portion of the grain silo is completely full, what is the total volume of grain?

6,925 ft^{3}

5,925 ft^{3}

4,925 ft^{3}

3,925 ft^{3}

**Answer:**

Since the dome shape will remain empty, the volume of the grain silo can be calculated using the formula \(V=πr^2h\). The radius of the silo is 5 feet and the height is 50 feet. When these two values are plugged into the formula, along with an approximation for pi (3.14), \(V=πr^2h\) becomes \(V=(3.14)(5)^2(50)\). This simplifies to 3,925. The silo contains 3,925 ft^{3} of grain.

**Question #4:**

Max makes candles to sell at his local farmers market. The candles that he makes are cylindrical. He bought a new candle mold, and he needs to figure out how much melted wax it can hold. The cylindrical mold has a height of 10 inches, and a radius of 5 inches. How much melted wax can he pour into the mold if he wants to fill it completely?

785 in^{3}

745 in^{3}

722 in^{3}

791 in^{3}

**Answer:**

To calculate the volume of wax that can be poured into the cylindrical mold, we can use the formula \(V=πr^2(h)\). We know that the height of the cylinder is 10 inches, and the radius is 5 inches. When we plug these into the formula, along with an approximation of π (3.14), \(V=πr^2(h)\) becomes \(V=(3.14)(5)^2(10)\), which simplifies to 785. The candle mold can hold 785 in^{3} of melted wax.

**Question #5:**

Julia wants to upcycle three old fruit cans. Each can is 5 inches tall and has a radius of 2 inches. She plans on painting the cans and using them as flower pots. If she only wants to paint the sides of the cylinders and the bottoms, what is the total surface area she will need to paint?

426.08 in^{2}

326.08 in^{2}

226.08 in^{2}

126.08^{2}

**Answer:**

In order to calculate the surface area of the three cans, we can make a few slight adjustments to the formula \(SA=2πr^2+2πrh\). Since Julia is only painting the sides of the cans \((2πrh)\), and the bottoms of the cans \((πr^2)\), we can adjust the formula so that we are adding three circles and three rectangles. The formula \(SA=2πr^2+2πrh\) becomes \(SA=3πr^2+3(2πrh)\). When we plug in 2 for the radius, 5 for the height, and 3.14 as an approximation for π, the formula becomes \(SA=3(3.14)(2)^2+3(2(3.14)(2)(5))\), which simplifies to 226.08. Julia will need to paint a total of 226.08 in^{2}.