Perimeter is the walk around the edge, area is the covering inside
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.
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.
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.
Symmetry and grids · Covering a Rectangle with L-Shaped Tiles
QuestionAn L-shaped tile is made of 3 unit squares joined edge to edge. A rectangle 6 units long and 4 units wide is covered with these tiles, with no gaps and no overlaps. (a) How many tiles are used? (b) What is the perimeter of one tile?
1.The rectangle is made of 6 × 4 = 24 unit squares.
The rectangle is 6 × 4 = 24 unit squares. 2.Each tile covers 3 unit squares, and the tiles cover the rectangle with no gaps and no overlaps, so the number of tiles is 24 ÷ 3 = 8.
Each tile covers 3 squares, with no gaps and no overlaps: 24 ÷ 3 = 8 tiles. 3.(a) 8 tiles are used. They really do fit: two tiles fit together to make a rectangle 3 units long and 2 units wide, and four of those fill the 6 by 4 rectangle.
(a) 8 tiles. Two make a 3 by 2 rectangle, and four of those fill the 6 by 4. 4.Now walk round the outside of one tile and count the unit edges: 2 along the bottom, 1 up, 1 back in, 1 more up, 1 across the top and 2 down the far side.
Walk round one tile: 2, 1, 1, 1, 1 and 2 unit edges. 5.(b) The perimeter of one tile is 2 + 1 + 1 + 1 + 1 + 2 = 8 units. Check: 3 separate squares have 3 × 4 = 12 edges, and each of the 2 joins hides 2 edges, one from each square, so 12 − 4 = 8.
(b) The perimeter of one tile is 2 + 1 + 1 + 1 + 1 + 2 = 8 units.
Answer: (a) 8 tiles; (b) 8 units
Watch out for
- Giving the perimeter of a tile as 12 units, the edges of 3 separate squares. Where two squares are joined, the shared edge is inside the tile, not on its outline, so each of the 2 joins takes away 2 edges: 12 − 4 = 8.
- Dividing the rectangle's perimeter by the tile's perimeter: 20 ÷ 8 is not even a whole number. The tiles cover the space inside the rectangle, so it is the areas that are divided: 24 ÷ 3 = 8 tiles.
Your turn
1.Area of a 5 by 3 rectangle