Video: Finding the Surface Area of a Composite Shape

The object shown is made from two cubes, one of side 3.3 cm, and the other 1.2 cm. If you must paint this object, what area do you report?

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Video Transcript

The object shown is made from two cubes, one of side 3.3 centimeters and the other 1.2 centimeters. If you must paint this object, what area do you report?

So in this question, in order to find the area, we need to calculate the surface area. And the surface area of a three-dimensional object is found by calculating the area of every face and adding those areas together. We can assume that the larger cube on the bottom has a length of side 3.3 centimeters and the smaller cube has a length of 1.2 centimeters. And since these are cubes, we know that every side will have the same length. So let’s begin by calculating the area of one of the faces on the larger cube.

Since each face on the cube is a square, we can calculate the area by multiplying the length by the length. And we substitute the values 3.3 by 3.3, which we can calculate as 10.89 and the units here will be centimeter squared. So, in the usual case of calculating the surface area of a cube, we could find the area of one face and then multiply it by six, since we know that there must be six equal areas. However, in this case, it’s slightly more complicated because there’s a smaller cube sitting on the top. So here, we have just five areas that would be the same, the one at the front and the back, the left and right, and on the base.

To find the area of the five sides, we multiply five by 10.89, which gives us 54.45 centimeter squared. So to find the area of the top of our large cube, we can visualize this as a square of side 3.3 centimeters minus the area of a square of 1.2 centimeters. In order to calculate this, we would find the area of our large square, subtract the area of the small square, which is equivalent to 3.3 times 3.3 subtract 1.2 times 1.2. This simplifies as 10.89 subtract 1.44. And we can simplify it as 9.45 centimeter squared.

Next, we need to calculate the area of one of the faces on our smaller cube. And since we found the area by multiplying the length times the length, this would be 1.2 times 1.2, which gives an area of one face as 1.44 centimeters squared. And in order to find the surface area of the smaller cube, we would usually multiply by six. But in this case, the base of our cube isn’t included in the calculation. So the surface area of five sides of the smaller cube is found by calculating five times 1.44 which is 7.2 centimeter squared.

And now, to find the total surface area, we need to pick out the key values the area of the five sides of our larger cube plus the area of the top of our larger cube plus the five sides of our smaller cube. And we add together 54.45, 9.45, and 7.2, giving us a final answer for the area that we must paint as 71.10 centimeter squared.

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