Video: Finding the Total Surface Area of a Cube given the Perimeter of Its Base

If the perimeter of the base of a cube is 54.4 cm, find its total surface area.


Video Transcript

If the perimeter of the base of a cube is 54.4 centimeters, find its total surface area.

The base of a cube is a square. Here’s the base, and the perimeter of this square is 54.4 centimeters. The distance around the base measures this much. We know that the base of a cube is a square and that in a square all sides are equal. If we add up all four of the sides, side plus side plus side plus side, it would be 54.4 centimeters.

In order to solve the problem though, we need to find out how much one side is worth. We have to divide 54.4 by four. How many times does four go into five? One time. One times four equals four. Five minus four equals one. Bring down the next digit. How many times does four go into 14? Three times. Three times four equals 12. 14 minus 12 equals two. Make sure we line up our decimal point for the quotient, and then bring down the next digit.

How many times can four go into 24? Six times. Six times four equals 24. 24 minus 24 equal zero. There is no remainder. This tells us that each of our sides measure 13.6 centimeters. We’ve only found the length of one side, but what is our goal here? Our goal is to find the total surface area of a cube. We know that the surface area of a cube is the area of all six faces added together. Let’s start by finding the area of the base.

Remember that the base is a square, and we know a side length. And to find the area of a square, we square its side length. Multiply side by side. We need to square 13.6. To do that, we multiply 13.6 times 13.6. Six times six is 36. Six times three is 18, plus the three we carried over equals 21. Write down the one; carry the two. Six times one is six, plus the two we carried over equals eight.

Now we’ll start multiplying with the three. Three times six equals 18. Write down your eight; carry your one. Three times three equals nine, plus one equals 10. Write down the zero; carry the one. Three times three is three, plus one is four. Now we’re multiplying by one. One times six equals six. One times three equals three. One times one equals one. From there, we add all the partial products.

Bring down the six. One plus eight equals nine. Eight plus zero plus six equals 14. Write down the four; carry the one. One plus four plus three equals eight. And then we bring down the one. We’re dealing with two decimal places. So we put our decimal two places to the left. And we’ve found the area of the base: 184.96 centimeters squared. Now that we know the area of one of the sides, we can find the total surface area.

A cube has six identical faces, and that means we can take the area of one of the faces and multiply it by six. To find the total surface area, 184.96 times six will give us our total surface area. Six times six equals 36. Six times nine is 54, plus three is 57. Six times four is 24, plus five is 29. Six times eight is 48, plus two is 50. Six times one is six, plus five is 11.

We see that there are two decimal places in our problem, and there will be two decimal places in the solution. By multiplying the area of one of the faces by six, we’ve found the total surface area to be 1109.76 centimeters squared.

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