How to Count the Square Area of an Irregular Shape

Calculating the exact area of an irregular shape can be difficult, depending on just how irregular its boundaries are. Shapes that are made up of a conglomeration of right-angle forms are actually quite easy to calculate exactly, but if your irregular shape has wavy or curved boundaries, things get a bit more difficult. Even if you can't calculate the shape's area without the benefit of calculus techniques, you can still get an estimate that's close to the exact answer.

Things You'll Need

• Graph paper
• Pencil

Instructions

1. Draw the shape out or copy it onto graph paper. If you're drawing a real world shape, such as the outline of a lake whose surface area you want to calculate, make sure that it's drawn to scale and that you know what the scale ratio is.

2. Draw a horizontal line through the shape, dividing it into a top and a bottom half, along one of the horizontal lines on the graph paper. Make four copies of this page. Black out the bottom half of the figure on two of the copies; on the other two copies, black out the top half of the figure but leave the bottom untouched.

3. Tackle the copies with the bottom half blacked out first. This means you're working on the top half of the figure only. Draw over the vertical graph paper lines just to the left and just to the right of the figure, tracing high enough to reach the horizontal line just above the highest point in the figure. Draw over that horizontal line, too. You should now have a box that contains the entire top half of the figure without any of the figure's outline "leaking" over the side.

4. Black out every square of graph paper that is inside the box you just drew, but outside the figure. Leave any squares of graph paper that have any portion of the figure's outline in them un-colored. Count the number of squares that are un-colored. This is your "upper estimate", or an estimate that will be just a bit bigger than the actual measurement. Note this on a separate piece of paper.

5. Repeat Step 3 with the other copy that has the bottom half of the figure blocked out. This time, instead of blacking out every square of graph paper that's inside the box but outside the figure, black out every square of graph paper that's completely inside the figure. If a square has any of the figure's outline in it, leave it blank. Count the number of squares that are colored in. This is your "lower estimate", or an estimate of the figure's area that is a bit smaller than the actual measurement. Note it on the separate paper as well. Keep in mind that so far you've only calculated upper and lower estimates for the top half of the figure; now you must follow the same procedure for the bottom half.

6. Repeat Steps 3 through 5 with the other copies, first drawing a box around the bottom half of the figure and then coloring the squares outside the figure but inside the box, as in Steps 3 and 4, to reach an "upper estimate". Note this on your paper. Then use the last copy, still working with the bottom half of the figure, to reach a "lower estimate" for the bottom half of the figure as described in Step 5.

7. Add all four figures you just calculated together, then divide that sum by two. This averages the upper and lower estimates of the figure, getting you reasonably close to the actual answer in between. If you need this number in terms of real world measurements, multiply the figure you calculated by the scale ratio of the figure you drew--so if your irregular shape's area averaged out to be 32 squares on the graph paper and when you drew the scale figure you determined that each square represented one foot in the real world, your figure's actual real world area would by 32 squares x 1 foot = 32 feet square.