How to Find the Area of an Octagon

This video shows how to find the are of an octagon. An octagon can be seen as a square with the corners cut off. The area of the octagon is the area of the square minus the area of the four missing corners, or triangles.

Area of an Octagon

Area of an Octagon

What is the area of an octagon with side length 8? An octagon is really just a square with the 4 corners cut off. If you found the area of the square, and then subtracted the area of the 4 triangles, you’d be left with the area of the octagon.

Let’s write down our plan. The area of the octagon is the area of the square, minus the area of the 4 triangles. I’m going to start by finding the area of the 4 triangles. Since we have a square here that means that this is a right angle, and we’d be cutting off equal sides here, (so this would be 45 degrees, 45 degrees, or it’s an isosceles triangle) and with side length 8, the hypotenuse of this right triangle is also a side of our octagon.

The hypotenuse of the right triangle is 8, and it’s not just any right triangle, it’s a 45-45-90 triangle, which means that the hypotenuse is the square root of 2, times the leg. Hypotenuse equals the square root of 2 times the leg.

If we want to find out what the legs are, so we can find the area of the triangle, then we can substitute 8 for hypotenuse, 8 is equal to the square to 2 times the leg, and solve for the leg. The opposite of multiplying times the square root of 2, is dividing by the square root of 2.

The square root of 2 divided by the square root of 2 is 1, so that the leg is 8 divided by the square root of 2. We don’t leave radicals in the denominator, so we’re going to rationalize the denominator by multiplying times the square root of 2, divided by the square root of 2.

The square root of 2 divided by the square root of 2 is, again, 1. Really, you’re not multiplying by anything. You’re really just rearranging the fractions so that your denominator doesn’t have a radical in it anymore.

To multiply fractions we multiply straight across, 8 times the square root of 2 is 8 square roots of 2, divided by the square root of 2 times the square root of 2 is the square root of 4, and the square root of 4 is 2. Then we would simplify, 8 square roots of 2 divided by 2 is 4 square roots of 2, so that means that all the legs are 4 square roots of 2.

That’s on all of our triangles, so I could even put it over here, and that will help us later to have that information. We can use that information now to find the area of one of our triangles. The area of a triangle is 1/2 the base times the height, where the base is any side of the triangle, but the height must be perpendicular to it.

Our perpendicular base and height would be our two legs, 4 square roots of 2, and 4 square roots of 2. The area of our triangle is 1/2 times 4 square roots of 2, times 4 square roots of 2, so the area is (you can do this really in any order, so let’s start with maybe the most complicated part) the square root of 2 times the square root of 2 is (again) the square root of 4, which is 2.

That’s 2, but then we’re taking 1/2 of it, so 1/2 of 2 is just 1, and 1 times 4, times 4 is 16, so 16 units squared. That’s for 1 triangle, but we have 4 triangles, so the area of the 4 triangles would be 4 times 16 units squared, which is 64 units squared.

We found this part now of how to find the area of our octagon, now we need to find the area of our square and then subtract those. The area of a square is side squared, and the length of one of my sides of my square is 4 square roots of 2 plus 8, plus 4 square roots of 2.

You can only add like terms, so we’ll add 4 square roots of 2 and 4 square roots of 2 to get 8 square roots of 2. The length of one of my sides is 8 plus 8 square roots of 2, and then that has to be squared to find the area of the square.

This is a binomial and that means to square it, I actually have to take it and multiply it times itself, so the area is 8 plus 8 square roots of 2, times 8, plus 8 square roots of 2. To multiply these binomials together we’ll use FOIL.

First term times the first term, 64, plus outer 8, times outer 8 square roots of 2. 8 times 8 is 64 square roots of 2. Inner, 8 square roots of 2 times 8, (again) 64 square roots of 2, and last, 8 square roots of 2 times 8 square roots of 2. 8 times 8 is 64, times the square root of 2, times the square root of 2, is again the square root of 4, which is 2 (so times 2).

Then we can simplify that, the area is 64 plus (I have some like terms here that I can add together) 64 square roots of 2, plus 64 square roots of 2 is 128 square roots of 2. Plus, and then 64 times 2 is 128. I still have like terms that I can combine, so I’ll do that on this line.

The area is 64 plus 128 is 192, plus 128 square roots of 2 units squared. Here we have the area of our square, and here we have the area of our 4 triangles, and to find the area of our octagon, now we need to subtract those. The area of the octagon is 192 plus 128 square roots of 2, minus 64, and that gives us 192 minus 64 is 128, plus 128 square roots of 2 units squared, and that’s the area of an octagon with side length 8.

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