To find the area of an irregular shape, trace it on graph paper and count the squares (full squares plus half the partial ones), or split it into rectangles and triangles, work out each area, and add them together.
Rectangles are easy. Puddles, rooms with angled walls and hand-drawn blobs need a few more tricks.
The area of a rectangle is length times width. Almost nothing you want to measure is a rectangle. Garden beds curve. Rooms have bay windows. A leaf, a lake on a map or a stain on the carpet has no formula at all.
You still have options. Here are five methods, from the one a school kid can do in five minutes to one that only needs a pencil and a kitchen scale.
| Method | Best for | What you need | Accuracy |
|---|---|---|---|
| Count the grid | Curvy shapes, drawings | Graph paper | Good, better with small squares |
| Split into pieces | Rooms, plots, straight edges | Tape measure | Very good |
| Shoelace formula | Any shape with straight sides | Corner coordinates | Exact |
| Pick's theorem | Shapes drawn on a dot grid | Dot or graph paper | Exact |
| Weigh the paper | Very wiggly outlines | Paper, scissors, fine scale | Good |
This is the classic, and it works for any shape you can trace.
Say you count 41 full squares and 18 partial ones. That gives 41 + 9 = 50 squares. If each square is 1 cm wide, the area is about 50 square centimeters.
The halving rule works because, on average, the partial squares are about half inside. Smaller squares make the guess better. Going from 1 cm squares to 5 mm squares means four times the counting, but the error shrinks a lot.
Most real spaces have straight edges. An L-shaped room is two rectangles. A plot with one slanted side is a rectangle plus a triangle.
A room 12 ft by 10 ft with a 4 ft by 3 ft closet nook is 120 + 12 = 132 square feet. That is how flooring is measured, and it is why installers ask for a sketch.
If you know the corners of a straight-sided shape as coordinates, this gives the exact answer. List the corners in order around the shape. Multiply each x by the next corner's y, and each y by the next corner's x. Subtract the two totals and halve the result.
Take a house shape with corners at (0,0), (6,0), (6,4), (3,7) and (0,4). The cross products are 0, 24, 30, 12 and 0. They add to 66, and half of that is 33.
Check it the long way: a 6 by 4 rectangle is 24, and the roof triangle is half of 6 times 3, which is 9. Total 33. The name comes from the crisscross pattern the multiplications make when you write them out.
If your shape has its corners on the dots of a grid, there is a neat shortcut. Count the dots strictly inside the shape (I) and the dots on its edges (B). Then:
Area = I + B/2 - 1
A shape with 10 inside dots and 8 edge dots has an area of 10 + 4 - 1 = 13 grid squares. It only works for straight-sided shapes on grid points, but for those it is exact.
This one feels like a magic trick. Paper of one kind has the same weight in every square inch, so weight tracks area.
A US Letter sheet is 8.5 by 11 inches, or 93.5 square inches. If the sheet weighs 4.6 g and your cutout weighs 1.2 g, the shape is about 93.5 x 1.2 / 4.6, or 24 square inches.
You need a scale that reads to 0.1 g or finer, like a small jewelry scale. Most kitchen scales only show whole grams, which is too rough for one sheet. Cardstock or a few stacked layers help.
The paper trick works because thin, flat things weigh in proportion to their area. A 2 inch wide strip weighs twice what a 1 inch strip of the same length does. Long thin bits count for very little.
That is the idea behind Halfsies, a free daily game where you cut three hand-drawn objects in half by eye and a scale weighs the two pieces. The weight comes from each half's drawn area, so a skinny handle barely moves the needle and a fat blade does. Your score for each cut is the lighter half divided by the heavier half, times 100, with 300 possible a day. The weigh-it method above is basically how the game judges you. You can try today's cuts in your browser, no download needed.
Trace it on graph paper and count the squares. Add the full squares to half of the partial squares, then multiply by the area of one square.
Split the floor plan into rectangles and triangles, measure each one, and add the areas. Subtract any cutouts like a chimney or stair opening.
Yes. If you know the corner coordinates, the shoelace formula gives the exact area of any shape with straight sides that do not cross.
Yes. Cut the shape from uniform paper, weigh it, and compare it with the weight of a sheet of known size. You need a scale that reads to 0.1 g.