Area and Perimeter (Composite Shapes)

Area is the amount of flat space a shape covers, measured in SQUARE units. Perimeter is the total distance around the outside of a shape, measured in plain units. In the US we measure rooms in feet and small objects in inches; in metric science work we use centimeters and meters. The formulas are exactly the same — only the unit labels change. Composite shapes are made of two or more rectangles joined together — once you split them apart, the work becomes ordinary rectangle math in whichever units the problem gives.

Area and Perimeter of a Rectangle

• Area of a rectangle = length x width. The answer is in SQUARE units: sq in, sq ft, sq yd (customary), or cm², m² (metric).
• Perimeter of a rectangle = 2 x (length + width), or just add all four sides: L + W + L + W. Same units as the sides (in, ft, cm, m).
• Customary example: a desktop 24 in by 18 in has area 24 x 18 = 432 sq in and perimeter 2 x (24 + 18) = 84 in.
• Metric example: a desktop 60 cm by 45 cm has area 60 x 45 = 2700 cm² and perimeter 2 x (60 + 45) = 210 cm.
• Area uses square units because you are filling 2-D space; perimeter uses plain units because you are walking around a 1-D path.
• Never mix systems in one problem: keep the whole calculation in inches OR cm, in feet OR meters.

What Is a Composite Shape?

• A composite shape is built from two or more simple shapes — usually rectangles — joined edge to edge.
• L-shapes, T-shapes, and step-shapes are all composite shapes you will see often.
• On the outline you can usually see places where two rectangles meet at a corner that points inward.

Finding Area by Decomposing

• Draw a straight line through the figure to split it into two rectangles that do not overlap.
• Label the length and width of each rectangle using the dimensions given on the figure.
• Compute the area of each rectangle (length x width).
• Add the rectangle areas together to get the total area of the composite shape.
• Customary L-shaped room: a 15 ft by 10 ft outer minus a 5 ft by 4 ft notch — or, split it: a 15 ft by 6 ft slab (90 sq ft) plus a 10 ft by 4 ft piece (40 sq ft) = 130 sq ft.
• Metric L-shaped room: a 4.5 m by 3 m outer minus a 1.5 m by 1.2 m notch — split into a 4.5 m by 1.8 m slab (8.1 m²) plus a 3 m by 1.2 m piece (3.6 m²) = 11.7 m².
• Example L-shape (unit-agnostic): a 10 by 3 bottom slab (area 30) plus a 4 by 5 vertical piece (area 20) gives 30 + 20 = 50 sq units.

Finding Perimeter by Walking the Outline

• Start at one corner and walk all the way around the outside of the shape.
• Add up every side you pass — including the inward-pointing edges of any notch.
• Do NOT add the dashed split line you drew inside the figure. That line is not part of the outline.
• Example L-shape sides: 10 + 3 + 6 + 5 + 4 + 8 = 36 units of perimeter.
• A useful check: opposite outer sides of an L-shape should still match up. The bottom (10) equals the top of the left vertical (4) plus the notch (6). The left side (8) equals the right side (3) plus the notch height (5).

Finding Missing Side Lengths

• Composite shapes often label only some sides — you must figure out the rest.
• For horizontal sides: the bottom edge equals the sum of every horizontal segment along the top of the figure.
• For vertical sides: the full left or right side equals the sum of every vertical segment on the opposite side.
• Example: if the bottom is 10 and one top piece is 4, the other top piece must be 10 - 4 = 6.