Hi Liz. The first thing to do is to draw a regular octagon. Thus, all sides of the 8 sided figure must be equal. Write a 4 for each side, because the perimeter of 32/8=4.
Then draw 2 parallel lines inside the figure. It will leave you with a rectangle in the middle, an isosceles trapezoid at top, and another one at the bottom.
Leave the rectangle untouched. But make small right triangles inside the octagon at the 2 top corners, and 2 bottom corners.
The hypotenuse (longest side) of each small right triangle has a length of 4, like I mentioned earlier. The 2 smaller sides each have a length of 2 x square root of 2. This is because the small triangles have angles of 45-45-90, so the 2 smaller sides are =. Also, use the Pythagorean Theorem to find the 2 square root of 2.
Then we calculate. The 4 small triangles total an area of 32 x square root of 2. That’s because each triangle’s area is 8 square root of 2. (We get that because we multiply the base x height, which is 2 square root of 2, times 4).
When we made the small triangles on top and bottom, we were left with 2 small rectangles on top and bottom. Using l x w to find these rectangles, we get 4x4 to get 16 for the top rectangle, and 4x4 to get 16 as the area of the bottom rectangle.
For the middle rectangle, remember it is larger in width. Adding 2 square root of two on both sides, to the 4 in the middle, gives you a width of 8 square root of 2. Now multiply that by that side of the octagon (4), and we get an area of 32 square root of 2.
The final answer is as added: 4 small triangles, 32 square root of 2
2 small rectangles, 32
Middle rectangle, 32 square root of 2
Total: 32 + 64 square root of 2
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