Lesson Guide
Explanation
If area is length times width, then area divided by one side is the other side. Rectangle P is ten by six, area sixty. Rectangle Q has the same area and is four wide, so its length is sixty divided by four, fifteen. Two steps when the area is not stated: find it from the rectangle you know, then divide by the known side of the other. Check by multiplying forwards: fifteen times four is sixty.
Your Turn
A rectangle has area 48 and width 6. Length? Check: 48 ÷ 6 = 8.
Watch Out
Divide the AREA by the side, not the side by the side. 60 ÷ 4 = 15, not 10 ÷ 4.
Explanation
Every rectangle with area twenty-four and whole-number sides is a factor pair of twenty-four: one by twenty-four, two by twelve, three by eight, four by six. Four rectangles, one area, and four different perimeters — fifty, twenty-eight, twenty-two, twenty. The squarer the rectangle, the smaller its perimeter, and the long thin one has the biggest. That is the factors lesson, drawn.
Your Turn
How many rectangles with whole-number sides have area 36? Check: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6 — five.
Watch Out
4 × 6 and 6 × 4 are the same rectangle turned round. Count each pair once.
Explanation
Area and perimeter are separate things, and a shape can change one while keeping the other. A four by six and a two by twelve both have area twenty-four, with perimeters twenty and twenty-eight. A one by nine and a five by five both have perimeter twenty, with areas nine and twenty-five. For a fixed area the square has the smallest perimeter, and for a fixed perimeter the square has the biggest area. When a question gives one and asks for the other, the sides always have to be found in between.
Your Turn
Which rectangle with whole-number sides and area 36 has the smallest perimeter? Check: 6 × 6, perimeter 24.
Watch Out
Same area does NOT mean same perimeter. Never carry a perimeter from one rectangle to another just because their areas match.
Explanation
Sides do not have to be whole numbers. A rectangle four long and one and a half high has area four times one and a half, six — four lots of one and a half. On a grid a half-unit side shows as a row of half-squares, and four half-squares make two whole ones. When a question shows several rectangles and asks which has a given area, multiply each one out; the size of the drawing can mislead, the numbers cannot.
Your Turn
Area of a 2½ by 4 rectangle, and of a 3 by 2½. Check: 10; 7½.
Watch Out
4 × 1½ is 6, not 4½ and not 5. Four lots of 1½: 1½ + 1½ + 1½ + 1½.
Explanation
Every question of this kind goes through the sides. Same area and one side known: divide. A rectangle reshaped into a square: the square's side is the number that multiplies by itself to give the area. Perimeter and area both given: the sides add to half the perimeter and multiply to the area — ten and twenty-four gives four and six. A square with area forty-nine has side seven and perimeter twenty-eight. Find the sides, and then whatever was asked is one more step. Quiz and exercises now.
Your Turn
A rectangle has perimeter 26 and area 40. What are its sides? Check: add to 13, multiply to 40 — 5 and 8.
Watch Out
Half the perimeter is the SUM of the two sides. Do not use the whole perimeter as the sum.
📌 Key Points — this lesson in a nutshell
- Area = length × width, so area ÷ one side = the other side. A rectangle of area 60 that is 4 wide is 15 long.
- Rectangles with the same area are the factor pairs of that area: 24 is 1 × 24, 2 × 12, 3 × 8, 4 × 6.
- Same area does not mean same perimeter. The squarer the rectangle, the smaller its perimeter.
- Sides can be halves: a rectangle 4 by 1½ has area 6. Multiply as with any numbers.
Step 1: Introduction
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