Can I use subtraction instead of addition to find the area?

Yes! Sometimes it is easier to imagine a larger regular shape, calculate its total area, and then subtract the area of the 'missing' pieces. For an L-shape, you could find the area of the large bounding rectangle and subtract the empty rectangular space.

What if the irregular shape has curved edges?

If the shape includes a curve, try to isolate it into a semi-circle or quarter-circle. Find its area using the circle formula $A = \pi r^2$ (remembering to divide by 2 or 4), and add or subtract it from the rest of the shape.

Do I get a different answer if I cut the shape differently?

No. As long as your pieces don't overlap, don't leave any gaps, and you calculate the new side lengths correctly, the total area will always be exactly the same regardless of how you decompose the shape.

How do you find the area of an irregular shape?

To find the area of an irregular shape, you break it down into smaller, standard shapes—like rectangles, triangles, or circles—calculate the area of each piece separately, and then add those areas together.

Imagine you have an oddly shaped floor that needs new carpet. Instead of trying to measure the whole thing at once with a complicated formula, you can visually split the room into a few standard rectangles. You calculate the carpet needed for each rectangle and add them up to cover the whole floor.

What is decomposition?

The method of breaking a complex shape into simpler ones is called decomposition. In geometry, there isn't a single formula for every possible irregular shape, so we rely on the area formulas we already know, like $A = l \times w$ for rectangles or $A = \frac{1}{2}bh$ for triangles.

Choosing the right cuts

You can often split an irregular shape in more than one way. The trick is to look for lines that create shapes with dimensions you already know or can easily figure out. Drawing dashed lines on your figure can help you visualize these cuts. For example, an L-shape can be cut vertically into a tall rectangle and a short one, or horizontally into a wide rectangle and a narrow one.

Where students slip up

A common mistake is using the original, full side length for one of your new, smaller pieces. When you cut a shape with a dashed line, you often divide an outer side into two smaller segments. Always double-check your new side lengths by subtracting the known parts from the total length before plugging numbers into your area formulas.

Worked through

Find the area of an L-shaped polygon. The total vertical height on the left is $10\text{ cm}$ , and the total horizontal base at the bottom is $8\text{ cm}$ . The top horizontal edge is $3\text{ cm}$ wide, and the rightmost vertical edge is $4\text{ cm}$ tall.

Step 1: Break the shape into two rectangles by drawing a vertical line down from the inside corner.

Step 2: Identify the dimensions of the left rectangle. It has a width of $3\text{ cm}$ and a height of $10\text{ cm}$ . Its area is $A_1 = 3 \times 10 = 30\text{ cm}^2$ .

Step 3: Identify the dimensions of the right rectangle. Its height is given as $4\text{ cm}$ . To find its width, subtract the top width from the total base: $8 - 3 = 5\text{ cm}$ . Its area is $A_2 = 5 \times 4 = 20\text{ cm}^2$ .

Step 4: Add the areas together. Total Area $= 30 + 20 = 50\text{ cm}^2$ .

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Where this comes from: OpenStax Prealgebra: Properties of Angles and Polygons · Khan Academy: Area and perimeter unit