How to find the surface area of a cube - ISEE Upper Level Quantitative Reasoning

We know that the volume of a cube can be found with the equation:

, where is the side length of the cube.

Now, if the volume is , then we know:

Either with your calculator or with careful math, you can solve by taking the cube-root of both sides. This gives you:

This means that each side of the cube is long; therefore, each face has an area of , or . Since there are sides to a cube, this means the total surface area is , or .

This question is very easy. Do not over-think it! All you need to do is calculate the area of one side of the cube. Then, multiply that by (since the cube has sides). Each side of a cube is, of course, a square; therefore, the area of one side of this cube is , or . This means that the whole cube has a surface area of or .

One of the faces of the cube could be drawn like this:

Notice that this makes a triangle.

This means that we can create a proportion for the sides. On the standard triangle, the non-hypotenuse sides are both , and the hypotenuse is . This will allow us to make the proportion:

Multiplying both sides by , you get:

To find the area of the square, you need to square this value:

Now, since there are sides to the cube, multiply this by to get the total surface area:

Recall that the formula for the surface area of a cube is:

, where is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side () by because a cube has equal sides.

For our data, we know that ; therefore, our equation is:

To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:

, where is the side length.

For our data, this gives us:

Taking the cube-root of both sides, we get:

Now, use the surface area formula to compute the total surface area:

, where is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side () by because a cube has equal sides.

For our data, this gives us:

To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:

, where is the side length.

For our data, this gives us:

Taking the cube-root of both sides, we get:

(You will need to use a calculator for this. If your calculator gives you something like . . . it is okay to round. This is just the nature of taking roots!).

Now, use the surface area formula to compute the total surface area:

, where is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side () by because a cube has equal sides.

For our data, this gives us:

Now, this could look like a difficult problem; however, think of the equation for finding the length of the diagonal of a cube. It is like the Pythagorean Theorem, just adding an additional dimension:

(It is very easy, because the three lengths are all the same: ).

So, we know this, then:

To solve, you can factor out an from the root on the right side of the equation:

Just by looking at this, you can tell that the answer is:

Now, use the surface area formula to compute the total surface area:

, where is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side () by because a cube has equal sides.

For our data, this is:

Now, this could look like a difficult problem. However, think of the equation for finding the length of the diagonal of a cube. It is like the Pythagorean Theorem, just adding an additional dimension:

(It is very easy, because the three lengths are all the same: ).

So, we know this, then:

To solve, you can factor out an from the root on the right side of the equation:

Just by looking at this, you can tell that the answer is:

Now, use the equation for the volume of a cube:

(It is like doing the area of a square, then adding another dimension!).

For our data, it is:

Substitute in the formula for the surface area of a cube:

To find the surface area of a cube, use the formula , where represents the length of the side. Since the side of the cube measures , we can substitute in for .

Your friend gives you a puzzle cube for your birthday. If the length of one edge is 5 cm, what is the surface area of the cube?

To find the surface area of a cube, use the following formula:

This works, because we have 6 sides, each of which has an area of

Plug in our known to get our answer:

A cube has a side length of , what is the surface area of the cube?

Surface area of a cube can be found as follows:

Plug in our side length to find our answer:

Making our answer:

One of your holiday gifts is wrapped in a cube-shaped box.

If one of the edges has a length of 6 inches, what is the surface area of the box?

We can find the surface area of a square by squaring the length of the side and then multiplying it by 6.

To find the surface area of a cube, we will use the following formula:

where a is the length of any side of the cube.

Now, we know the width of the cube is 6cm. Because it is a cube, all sides are 6cm. That is why we can choose any side to substitute into the formula.

Now, knowing this, we can substitute into the formula. We get

To find the surface area of a cube, we will use the following formula:

where a is the length of any side of the cube. Note that all sides are equal on a cube. That is why we can use any length in the formula.

Now, we know the length of the cube is 7in.

Knowing this, we can substitute into the formula. We get

To find the surface area of a cube, we will use the following formula.

where a is the length of any side of the cube.

Now, we know the width of the cube is 4in. Because it is a cube, all sides are equal (this is why we can use any length in the formula). So, we will use 4in in the formula. We get

To find the surface area of a cube, we will use the following formula:

where l is the length, and w is the width of the cube.

Now, we know the length of the cube is 12in. Because it is a cube, all lengths, widths, and heights are the same. Therefore, the width is also 12in.

Knowing this, we can substitute into the formula. We get

While exploring an ancient ruin, you discover a small puzzle cube. You measure the side length to be . Find the cube's surface area.

To find the surface area, use the following formula:

A cube has all equal sides (length, width, height). To find the surface area of a cube, we will use the following formula:

Now, we know the base of the cube is 8in. Because all sides are equal (i.e. all sides are 8in), we will use that to substitute into the formula. We get