FG14–FG15, HG16–HG17 - Perimeter and polygon areas

Perimeters

A rectangle is $12\,\mathrm{cm}$ wide and $9\,\mathrm{cm}$ high. A $5\,\mathrm{cm}$ by $4\,\mathrm{cm}$ rectangle is cut from its top-right corner. Does removing that corner make the perimeter smaller?

Opposite sides of a rectangle are equal. For rectilinear figures, horizontal lengths can be added or subtracted to find other horizontal lengths; do the same separately for vertical lengths. A line used to split the interior is not part of the outer boundary.

Keep the outer dimensions, but remove the stated corner. The boundary now has six edges.

[DIAGRAM: asset_name: Perimeter of a corner-cut rectangle; asset_slug: corner-cut-rectangle; caption: Match horizontal lengths with horizontal lengths and vertical lengths with vertical lengths.; description: Exact L-shaped polygon from a 12 cm by 9 cm rectangle with a 5 cm wide, 4 cm deep top-right corner removed. Boundary labels show bottom 12 cm, left 9 cm, top 7 cm and lower right 5 cm; the two notch edges are 5 cm and 4 cm.]

This conclusion applies to this rectangular corner cut. A hole or a curved cut can change the boundary differently; trace the actual edges rather than assume that removing material always reduces perimeter.

Try a new corner cut.

Area formulae

Area counts how many unit squares cover a flat region, so it uses squared units. A $4\,\mathrm{cm}$ by $3\,\mathrm{cm}$ rectangle contains three rows of four unit squares: $4\times3=12\,\mathrm{cm}^2$.

A perpendicular height is measured at right angles to the chosen base. It is the separation between parallel lines containing the base and the opposite edge or vertex. A sloping side is usually a different length.

[DIAGRAM: asset_name: Polygon bases and perpendicular heights; asset_slug: polygon-area-heights; caption: The height meets the base direction at a right angle; a sloping side is not the height.; description: Four exact labelled panels show rectangle length and width, triangle base and perpendicular height, parallelogram base and perpendicular separation, and trapezium parallel sides a and b with perpendicular height h. Right-angle markers distinguish heights from sloping edges.]

Cutting a triangle from one end of a parallelogram and moving it to the other makes a rectangle of the same base and height, so its area is $bh$. Two identical triangles can make a parallelogram, so one triangle has half that area. Two identical trapezia can make a parallelogram with base $a+b$ and height $h$, giving half of $(a+b)h$ for each trapezium.