Perimeter of Compound Shapes

What is a compound shape?

A compound shape (also called a composite shape) is built by joining simpler shapes — usually rectangles — edge to edge. An L-shape, a T-shape or a staircase outline are all compound shapes.

The perimeter is still just the total distance once around the outside edge. The challenge is that some side lengths are often missing, so you must work them out first.

Step 1: walk around the outline

The safest method is to start at one corner and travel all the way round, writing down each edge as you pass it, until you return to where you began. This guarantees you count every outside edge once and none twice.

Described diagram: picture an L-shape. The full bottom edge runs 9 cm to the right. The right edge goes up 4 cm. Then the outline steps left 5 cm, up 3 cm, left 4 cm (back to the left edge), and finally down 7 cm to the start. Each arrow is one edge of the boundary.

Step 2: find the missing lengths

Often one or two edges are unlabelled. Two simple rules unlock them:

• Horizontal balance: the horizontal edges pointing right must total the same length as those pointing left.
• Vertical balance: the vertical edges pointing up must total the same as those pointing down.

Example: an L-shape has a full bottom of 9 cm. Higher up, one horizontal step is 5 cm. The hidden short step must make the two sides balance: 9 − 5 = 4 cm.

For the vertical sides, suppose the tall left edge is 7 cm and one short right edge is 4 cm. The remaining vertical step is 7 − 4 = 3 cm.

Worked example

An L-shaped patio has these outside edges, found using the balancing rules:

9 cm (bottom), 4 cm (up), 5 cm (left), 3 cm (up), 4 cm (left), 7 cm (down)

Add them all:

• 9 + 4 + 5 + 3 + 4 + 7 = 32 cm

That total is the perimeter. Always include the little cm (or other unit) — perimeter is a length.

The most common mistake

Do not include any line drawn inside the shape. If you split an L-shape into two rectangles to help you think, the dividing line is imaginary — it is not part of the boundary, so it never goes into the perimeter. Perimeter follows the outside edge only.

A second method: perimeter is unchanged by steps

Here is a useful shortcut for rectilinear shapes (where every corner is a right angle). The total of all the horizontal edges equals twice the overall width, and the total of all the vertical edges equals twice the overall height — no matter how many steps there are. So for an L-shape that fits inside a 9 cm by 7 cm box:

• Perimeter = 2 × (9 + 7) = 2 × 16 = 32 cm

This matches the walk-around answer, which is a great way to check your work.

Activity: design a robot

On squared paper, draw a "robot" outline using only horizontal and vertical lines and right angles. Label some edges and leave two blank. Swap with a partner, find the missing lengths using the balancing rules, then add up the full perimeter. Check it with the box shortcut.

Where this connects

This extends the boundary ideas from area and perimeter. Once you can find the perimeter of a compound shape, you can also split it into rectangles to find its area, which links neatly to types of quadrilaterals.