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How to find the Surface Area

If you think that the Surface Area is similar to the area of a shape, you're right on track! Except, when finding the surface area, you need to account for all the sides that are facing outwards (we're dealing with 3D shapes!) So when you have a cube, you find the area of one side of it, then multiply that by all the external sides, and that's it! Let's look at all the popular shapes...

Finding the Surface Area of a Cube
As we touched on above, a cube is the same thing as a square, but with more sides. Since we already know how to get the area of a 2D rectangle, all we need to do is get that area, and multiply by the number of sides pointing outward. Since a cube has 6 sides, we find the area of 1 side, and then multiply by 6!

Formula: Surface Area of a Cube = 6 * Area of 1 Side (This works for a perfect square only.)
Alternate Formula: Area of Cube = Area of each unique side * 2. (As shown below.)
Another way of stating the alternate formula would be: 2 * Area of Top (or Bottom) Rectangle + 2 * Area of Left (or Right) Rectangle + 2 * Area of Front (or Back) Rectangle

Since the above 3D Rectangle has various different size sides, we compute the area of the TOP (3*4=15), one SIDE (3*4=12), and the FRONT (5*4=20). So we have 15, 12, and 20. Since there is another side that is equal to what we've found, we multiply each of those numbers by two. (The top and bottom are the same size, both sides are the same size, and the front and back are the same size.) This gives us 30, 24, and 40. Add these up, and you get the answer of 94!

Finding the Surface Area of a Sphere
This one is pretty easy! It's is the same as the area of a circle, multiplied by 4!

Formula: Surface Area of a Sphere = 4 * pi * radius²

Using our sphere of radius 4 inches, we calculate 4 * 3.14 * 4². This is 4 * 3.14 * 16, giving us a surface area of 200.96 square inches!


Finding the Surface Area of a Cylinder
This one requires a couple of steps. We need to find the area of the top and bottom of the cylinder, as well as the area up and down the actual cylinder.

First, the surface area for the top and bottom of the cylinder: 2 * pi * radius²

Then we add this to the surface area of the rest of the cylinder: 2 * pi * radius * height

Formula: Surface Area of Cylinder = (2*pi*radius²) + (2*pi*radius*height)

First, we need to calculate the surface area of the top (and bottom) of the cylinder: 2 * 3.14 * 2², which totals 25.12. Then, we get the rest of the cylinders surface area: 2 * 3.14 * 3 * 5, which gives us 94.2. Add the two totals together to get the overall surface area: 25.12 + 94.2 = 119.32.

Finding surface area of 3D shapes is often the least-focused topic in geometry, compared to area and volume. However, surface area is arguable equally if not more important!

And there we have it, we're all done! Head over and review more Math Topics, or select a different subject at our Lesson Center!


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