Perimeter and Area

Perimeter: the distance around

The perimeter of a shape is the total distance around its edge — just walk around the outside, adding up the side lengths as you go.

For a rectangle with length l and width w, opposite sides are equal, so there are two lengths and two widths:

P = 2(l + w)

For any shape, there is no shortcut — just add every side length together. A triangle with sides 3, 4 and 5 has perimeter 3 + 4 + 5 = 12.

Area: the space inside

The area of a shape is how much flat space it covers, measured in square units — little 1 \times 1 squares that tile the inside.

A rectangle is a neat grid of those squares: l columns and w rows, so you multiply to count them all. A triangle is exactly half of the rectangle that boxes it in, so its area is half of base times height:

A_{\text{rect}} = l \times w \qquad A_{\text{tri}} = \tfrac{1}{2} \times b \times h

For a rectangle of length l and width w, and a triangle of base b and height h:

perimeter of the rectangle is 2(l + w) — for any shape, add up its side lengths;
area of the rectangle is l \times w, counted in unit squares;
area of the triangle is \tfrac{1}{2} \times b \times h.

Seeing the area

Step through the figure: first a rectangle filled with its unit-square grid, then a triangle with its base and (dashed) height marked.