Area of compound shapes

Composite shapes are rectangles in disguise. An L-shape is two rectangles sharing an edge, and a T-shape is a band plus a stem. Find each rectangle with length times width, then add the areas. A 3 by 5 block plus a 2 by 4 block gives 15 + 8 = 23. Adding is the whole trick. Never multiply the two block areas together and never add side lengths as if they were areas.

Grids prove splits work. Draw the L on squared paper, cut along the split line, and count each chunk. The counts add to the same total whatever split you chose. Two different splits of one shape must agree: 23 and 23 means solved, 23 and 21 means one split leaked. Overlaps are the classic leak. Chunks must meet edge to edge with no double covered squares. Touching is fine, overlapping double counts.

Floor plans turn splits into money and materials. A T floor with a 10 square metre wing and a 6 square metre wing needs 16 square metres of flooring. Tiles, paint and carpet are all ordered from the total, so an error multiplies across the bill. Real plans mix harder lengths, but the method never changes. Split into rectangles, area each, add. Write the unit on every chunk so metres never sneak in among square metres.

Area hunters carry many tools, and splitting composites is the sharpest. Rectangles fall to length times width. L-shapes and T-shapes fall to split, area, add. Triangles and circles need their own formulas, but they still add the same way into totals. Pick the tool per chunk: rectangle rule for boxy bits, special rules for the rest. Finish by asking if the total looks sane next to a rough box round the whole shape.