Rectilinear shapes are polygons with all right angles — L-shapes, T-shapes, and other stepped figures. Their area is found by decomposing (breaking apart) the shape into non-overlapping rectangles, finding each rectangle's area, and adding the parts together. Alternatively, students can find the area of a larger containing rectangle and subtract the missing piece.
Draw dashed lines on L-shapes to show the decomposition. Have students label the dimensions of each resulting rectangle, calculate each area, and add. Comparing the 'add parts' and 'subtract missing piece' strategies builds flexibility.
You already know two things: how to find the area of a rectangle using length × width, and that area counts the unit squares covering a surface. Now you will combine those skills to tackle shapes that aren't simple rectangles — the L-shapes, T-shapes, and stepped figures called rectilinear shapes. The key insight is that every rectilinear shape is secretly several rectangles stuck together.
Take an L-shape. It looks like a single irregular polygon, but draw one dashed line in the right place and it splits cleanly into two rectangles. Find each rectangle's dimensions, compute each area (length × width), and add the two results. The total area equals the sum of its parts. The tricky step is determining missing side lengths — the dimensions that aren't directly labeled on the figure. Because all angles in a rectilinear shape are right angles, every missing length can be calculated by looking at what's on the opposite side: sides facing each other along a straight line must add up to the same total.
A second strategy sometimes saves work: subtraction. Imagine drawing the smallest possible rectangle that completely surrounds the L-shape — a large rectangle with a rectangular notch cut out of one corner. Calculate the big rectangle's area, then calculate the notch's area, and subtract. Both the "add the pieces" method and the "subtract the missing chunk" method always produce the same answer. Choosing between them is just a matter of which side lengths are easier to work with on a given problem.
The deeper idea here is decomposition: when a shape seems too complicated, break it into simpler pieces you already know how to handle, solve each piece, then recombine. This strategy — reduce a hard problem to a collection of easy ones — extends far beyond geometry. It appears in every branch of mathematics and is one of the most important problem-solving habits you can build.