The walk around the edge
The perimeter of a shape is the total distance around its edge. Imagine walking all the way around a rectangle 5 cm long and 3 cm wide until you are back where you started. You walk 5 cm, then 3 cm, then 5 cm, then 3 cm, so the perimeter is 5 + 3 + 5 + 3 = 16 cm.
A perimeter is a length, so it is measured in units of length, such as centimeters (cm) or meters (m).
The rectangle has two sides of 5 cm and two sides of 3 cm. Its perimeter is 5 + 3 + 5 + 3 = 16 cm.
The covering inside
The area of a shape is the amount of flat surface inside it. To measure it, cover the shape with squares 1 cm long and 1 cm wide, called centimeter squares, and count them. The 5 by 3 rectangle holds 3 rows of 5 squares, so its area is 5 × 3 = 15 square centimeters, written 15 cm².
Area counts squares, so it is measured in square units, such as cm² or . A perimeter given in cm², or an area given in cm, is a sign that the two have been mixed up.
3 rows of 5 centimeter squares cover the rectangle: 5 × 3 = 15 cm².
Cut a corner out
Cut a 2 cm by 2 cm square out of one corner of a rectangle 7 cm long and 5 cm wide. The area gets smaller. The rectangle held 7 × 5 = 35 squares, the cut takes away 2 × 2 = 4 of them, and 31 cm² is left.
The perimeter does not get smaller. Where the corner was, the walk now turns in: it goes down 2 cm into the gap and across 2 cm out of it. Those are the same two lengths the cut took off the top and the side, so the walk is still 7 + 5 + 7 + 5 = 24 cm.
The walk around the 7 × 5 rectangle is 7 + 5 + 7 + 5 = 24 cm, and it covers 7 × 5 = 35 cm²
Cut a 2 × 2 corner out of the 7 × 5 rectangle, then walk the edge
Drag the two handles to cut a 2 by 2 corner out of the 7 by 5 rectangle. The two new edges are the same lengths as the two dashed edges the cut removed, so the perimeter stays 24 cm while the area loses 4 squares.
Same perimeter, different area
Area and perimeter measure different things, so knowing one does not tell you the other. A 4 by 4 square and a 1 by 7 rectangle both have a perimeter of 16: 4 + 4 + 4 + 4 = 16, and 1 + 7 + 1 + 7 = 16. But the square covers 4 × 4 = 16 squares, and the long thin rectangle covers only 1 × 7 = 7.
a = 4: the square, and the largest area 16 that 16 units of fence can hold
Stretch the rectangle into a 1 × 7 strip with the same fence
Every rectangle here has a perimeter of 16. Drag the side to stretch the square into a 1 by 7 strip: the perimeter stays 16, and the area falls from 16 to 7.
Add for the perimeter, multiply for the area
For a rectangle, add all four sides to find the perimeter, and multiply the length by the width to find the area. The usual slips mix the two: 5 × 3 = 15 is the area of the 5 by 3 rectangle, not its perimeter. And 5 + 3 = 8 is neither, because it walks only two of the four sides.
Worked example: Tiling with L-Shaped Pieces
Question An L-shaped tile is made of three unit squares. A rectangle 6 units by 4 units is covered completely with these tiles, none overlapping. How many tiles are used, and what is the perimeter of one tile?
1.Rectangle = 6 × 4 = 24 unit squares; each tile is 3: 24 ÷ 3 = 8 tiles.
24 unit squares to cover, 3 per tile: 8 tiles. 2.Two L-tiles fit together to make a 2 × 3 rectangle, and four such rectangles tile the 6 × 4: so 8 tiles really do fit.
Two L-tiles make a 2 by 3 block, and four blocks fill the 6 by 4. 3.One L-tile: three squares have 12 edges; two pairs of squares share an edge, hiding 2 × 2 = 4 of them.
Three squares have 12 edges; the two joins hide 4. 4.Perimeter = 12 − 4 = 8 units.
Perimeter of one tile: 12 − 4 = 8 units.
Answer: 8 tiles; 8 units
Common mistakes
- Giving the tile's perimeter as 12, counting the hidden shared edges.
- Dividing by 4, the number of squares in a 2 × 2 block, instead of the tile's 3.