Math - How to find the surface area of a cube

If the surface area of a cube equals 96, what is the length of one side of the cube?

Explanation

The surface area of a cube = 6a2 where a is the length of the side of each edge of the cube. Put another way, since all sides of a cube are equal, a is just the lenght of one side of a cube.

We have 96 = 6a2 → a2 = 16, so that's the area of one face of the cube.

Solving we get √16, so a = 4

$\textup{What is the surface area, in terms of }p,\textup{of a cube with sides of length }p\textup{ ?}$

$6p^{2}$

$p^{2}$

$4p^{2}$

$p^{3}$

Explanation

$\textup{Each of the six identical faces of the cube is a square with sides }p.$

$\textup{Each face: }A=p^{2}: : : : : \textup{Surface area}=6p^{2}$

This figure is a cube with one face having an area of 16 in2. Cube

What is the surface area of the cube (in2)?

Explanation

The surface area of a cube is the sum of the area of each face. Since there are 6 faces on a cube, the surface area of the entire cube is $6\cdot16$ .

What is the surface area of a cube with a side length of ?

$311in^{2}$

$373in^{2}$

$331in^{2}$

$52in^{2}$

$86in^{2}$

Explanation

In order to find the surface area of a cube, use the formula $SA=6(s^{2})$ .

$SA=6*(7.2in)^{2}$

$SA=6*51.84in^{2}$

$SA=311.04in^{2}$

$\rightarrow 311in^{2}$

The side length of a particular cube is $\frac{3}{2}$ . What is the surface area of this cube?

$\frac{27}{2}$

$\frac{9}{2}$

Explanation

To find the surface of a cube, use the standard equation:

where denotes the side length.

Plug in the given value for to find the answer:

$SA=6\cdot \left(\frac{3}{2}\right)^2=6\cdot \left(\frac{9}{4}\right)=\frac{27}{2}$

Find the surface area of the following cube:

Length_of_diagonal

$96\ m^2$

$90\ m^2$

$88\ m^2$

$82\ m^2$

$80\ m^2$

Explanation

The formula for the surface area of a cube is

where is the length of the side.

Plugging in our values, we get:

$SA=6(4\ m)^2$

$SA=96\ m^2$

Square_with_diagonal What is the surface area of a cube with a diagonal of $\dpi{100} d=4.2cm$ ?

$52.92cm^{2}$

$59.22cm^{2}$

$105.84cm^{2}$

$84.05cm^{2}$

$42.22cm^{2}$

Explanation

A few facts need to be known to solve this problem. Observe that the diagonal of the square face of the cube cuts it into two right isosceles triangles; therefore, the length of a side of the square to its diagonal is the same as an isosceles right triangle's leg to its hypotenuse: $1:\sqrt{2}$ .

$\frac{s}{d}=\frac{1}{\sqrt{2}}$

$\dpi{100} \dpi{100} \frac{s}{4.2cm}=\frac{1}{\sqrt{2}}$

Rearrange an solve for .

$\dpi{100} s=\frac{4.2cm}{\sqrt{2}}$

Now, solve for the area of the cube using the formula $\dpi{100} \dpi{100} SA=6(s^{2})$ .

$SA=6*(\frac{4.2cm}{\sqrt{2}})^{2}$

$SA=\frac{6}{1}*(\frac{4.2cm}{\sqrt{2}}*\frac{4.2cm}{\sqrt{2}})$

$\dpi{100} SA=\frac{6}{1}*(\frac{17.64cm^{2}}{2})$

$\dpi{100} \dpi{100} SA=3*17.64cm^{2}$

$\rightarrow 52.92cm^{2}$

A sphere with a volume of $\frac{32}{3}$ $\pi m^{3}$ is inscribed in a cube, as shown in the diagram below.

Act4

What is the surface area of the cube, in $m^{2}$ ?

$24\ m^{2}$

$48\ m^{2}$

$48\pi ^{2}\ m^{2}$

$48\pi \ m^{2}$

$96\ m^{2}$

Explanation

We must first find the radius of the sphere in order to solve this problem. Since we already know the volume, we will use the volume formula to do this.

$V_{sphere}=\frac{4}{3}\pi r^{3}$

$\frac{32}{3}\pi =\frac{4}{3}\pi r^{3}$

$\frac{32}{3}=\frac{4}{3}r^{3}$

$\frac{3}{4}\cdot \frac{32}{3}= r^{3}$

$8=r^{3}$

With the radius of the sphere in hand, we can now apply it to the cube. The radius of the sphere is half the distance from the top to the bottom of the cube (or half the distance from one side to another). Therefore, the radius represents half of a side length of a square. So in this case

$side=2\cdot 2=4$

The formula for the surface area of a cube is:

$SA_{cube}=6s^{2}$

$6s^{2}=6\cdot (4)^{2}=6\cdot 16=96$

The surface area of the cube is $96\ m^{2}$

Sarah is wrapping a birthday present. The box is a cube with sides of . At a minimum, how many square feet of wrapping paper will she need?

$1.50; ft^{2}$

$0.50; ft^{2}$

$1.25; ft^{2}$

$0.75; ft^{2}$

$1.00; ft^{2}$

Explanation

Remember, .

For a cube:

$SA = 6s^{2}$

Thus $6(0.50)^{2}=6(0.25)=1.50; ft^{2}$ .

What is the surface area of a cube with a side length of ?

Explanation

To find the surface area of a cube, we must count the number of surface faces and add the areas of each together. In a cube there are faces, each a square with the same side lengths. In this example the side length is .