The cross-section of a metal bracket is drawn on a square grid. The side of each small square represents \(1\) cm. (a) Find the area of the cross-section. [...

This question tests area and perimeter of a shape made of whole grid squares, where the shape is not a simple rectangle.

(a) Each small square has side \(1\) cm and so represents \(1\) cm\(^2\). Counting the squares that make up the cross-section gives

\[ 33 \text{ cm}^2 \] [B1]

(b) The perimeter is the total distance around the outside, so add the lengths of all the edges of the cross-section as you travel once around the boundary. [M1]

Because every edge of this shape runs along a grid line, the horizontal edges together travel the full width twice and the vertical edges together travel the full height twice. With a width of \(8\) cm and a height of \(6\) cm this gives

\[ 2 \times (8 + 6) = 28 \text{ cm} \] [A1]

Notice what this shows: the bracket has a smaller area than the \(8 \times 6 = 48\) cm\(^2\) rectangle that just contains it, but exactly the same perimeter, because each step cut into the shape removes area while simply redistributing the same total horizontal and vertical travel. This shortcut only works while every notch is a rectangular step; if the boundary had a sloping or curved edge, you would have to add the individual edge lengths one by one.