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

The length of the side of a cube is . Give its surface area in terms of .

$6x^{2} -60x + 150$

$6x^{2} -10x + 25$

$6x^{2} -60x - 150$

Explanation

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

$A = 6s^{2}$

$A = 6 \left ( x - 5\right )^{2}$

$A = 6 \left ( x^{2} - 2 \cdot 5 \cdot x + 5^{2}\right )$

$A = 6 \left ( x^{2} - 10x +25\right )$

$A = 6 x^{2} - 60x +150$

If a cube has one side measuring cm, what is the surface area of the cube?

Explanation

To find the surface area of a cube, use the formula $6s^{2}$ , where represents the length of the side. Since the side of the cube measures , we can substitute in for .

$6(4)^{2}=96: cm^{2}$

Find the surface area of a cube with a length of 9cm.

$486\text{cm}^2$

$324\text{cm}^2$

$218\text{cm}^2$

$576\text{cm}^2$

$338\text{cm}^2$

Explanation

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

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

Now, we know the length of the cube is 9cm. So, we get

$SA = 6 \cdot (9\text{cm})^2$

$SA = 6 \cdot 81\text{cm}^2$

$SA = 486\text{cm}^2$

Find the surface area of a cube with a width of 6in.

$216\text{in}^2$

$27\text{in}^2$

$144\text{in}^2$

$72\text{in}^2$

$432\text{in}^2$

Explanation

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

$SA = 6 \cdot l \cdot w$

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

Now, we know the width of the cube is 6in. Because it is a cube, all lengths, widths, and heights are the same. Therefore, the length is also 6in.

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

$SA = 6 \cdot 6\text{in} \cdot 6\text{in}$

$SA = 6 \cdot 36\text{in}^2$

$SA = 216\text{in}^2$

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

Explanation

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:

$SA_{cube}=6s^2$

Plug in our side length to find our answer:

$SA_{cube}=6(9mm)^2=6*81mm^2=486mm^2$

Making our answer:

Find the surface area of a cube with a width of 7in.

$294\text{in}^2$

$343\text{in}^2$

$84\text{in}^2$

$147\text{in}^2$

$245\text{in}^2$

Explanation

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 7in. Because it is a cube, all sides are equal (this is why we can use any length in the formula). So, we will use 7in in the formula. We get

$SA = 6 \cdot (7\text{in})^2$

$SA = 6 \cdot 49\text{in}^2$

$SA = 294\text{in}^2$

Find the surface area of a cube with a width of 4cm.

$96\text{cm}^2$

$64\text{cm}^2$

$48\text{cm}^2$

$24\text{cm}^2$

$128\text{cm}^2$

Explanation

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

$SA = 6 \cdot (4\text{in})^2$

$SA = 6 \cdot 16\text{in}^2$

$SA = 96\text{in}^2$

You are building a box to hold your collection of rare rocks. You want to build a cube-shaped box with a side length of 3 feet. If you do so, what will the surface area of your box be?

Explanation

You are building a box to hold your collection of rare rocks. You want to build a cube-shaped box with a side length of 3 feet. If you do so, what will the surface area of your box be?

To find the surface area, use the formula:

Where s is our side length and SA is our surface area.

Plug in our known and simplify

So, our answer is

Find the surface area of a cube with a length of 7in.

$294\text{in}^2$

$343\text{in}^2$

$49\text{in}^2$

$649\text{in}^2$

$412\text{in}^2$

Explanation

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

$SA = 6 \cdot (7\text{in})^2$

$SA = 6 \cdot 49\text{in}^2$

$SA = 294\text{in}^2$

What is the surface area of a cube with a volume ?

Explanation

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:

How to find the surface area of a cube

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