Area of compound shapes
Recognise area as additive; find areas of rectilinear figures by decomposing into non-overlapping rectangles and summing their areas
What a learner can do afterwards
- Decompose an L-shape into two rectangles, find each area, and add them
- Find the area of a floor plan shaped like a T by splitting into rectangles
- Solve: A room is L-shaped (3m×5m plus 2m×4m). What is the total area?
1 · Read
A compound shape is made of two or more rectangles joined together, like an L-shape or a T-shape. You already know that area is the amount of space inside a shape, and area is additive: you can split a shape into parts and add up their areas to get the whole.
A room is L-shaped. Draw one straight line to split it into two rectangles: one is 3m by 5m, the other is 2m by 4m. First rectangle: 3 x 5 = 15 square meters. Second rectangle: 2 x 4 = 8 square meters. Total area: 15 + 8 = 23 square meters.
Draw your split line first, then label the sides of each rectangle before you multiply. Check that your rectangles do not overlap and together cover the whole shape, or your total will be wrong.
Split a compound shape into rectangles, find each rectangle's area, then add them together for the total.
More to read
Key ideas: reason to find the area of composite shapes by using rectangles
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.
Reason to find the area of composite shapes by using grids
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.
Key ideas: Solve real-world problems involving areas of composite figures with mixed-number side lengths
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.
10 Ways to Find Area
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.
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18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.