How to calculate the square footage of an octagon
A regular octagon is an eight-sided shape where every side and every angle is identical — think of a stop sign or a classic garden gazebo. Because the shape is perfectly symmetrical, its area follows a single tidy formula: 2(1 + √2) × a², where a is the length of one side. That constant, 2(1 + √2), is roughly 4.828, so in practice you just square the side length and multiply by 4.828. A gazebo with 5 ft sides covers 4.828 × 25 ≈ 120.7 square feet.
The one thing to get right is which measurement you have. If you measured a single edge, use the side formula. If you measured the width straight across the middle — flat-to-flat between the two parallel sides — use area ≈ 0.828 × w² instead. Mixing them up is the most common mistake, because the flat-to-flat width of a regular octagon is about 2.414 times the side length, so the two numbers are far apart. This shape is closely related to the hexagon, which uses its own six-sided constant.
Splitting an octagon into simpler shapes
If you'd rather not trust a single constant, you can break the octagon into pieces you already know how to measure. The easiest approach is to picture the octagon as a central rectangle (actually a square in a regular octagon) with four right triangles filling the corners. Measure the middle square, then add the four triangles using the triangle tab in the calculator above. Add all five pieces together for the total.
The mirror-image method works too: enclose the whole octagon inside one big square, measure that square, then subtract the four right triangles that get cut off at each corner. Both methods give the same answer as the 4.828 formula, and they're handy when your octagon isn't perfectly regular. Unlike a rounded shape such as a circle, an octagon divides cleanly into straight-edged rectangles and triangles.
Turning square footage into a material estimate
Once you know the area, ordering material is straightforward. For a gazebo floor or an octagon deck, take the square footage and add a small waste allowance for cuts around the angled edges — the eight 45-degree corners generate more offcuts than a plain rectangle. Multiply the final figure by your price per square foot to budget for decking, tile, or pavers, and round up when buying by the box or slab so you don't come up short mid-project.