HSPT: Comparisons: area of rectangles and squares
Learn to compare the area of a rectangle with the area of a square by computing both, not by eyeballing side lengths.
How do you decide whether a rectangle or a square covers more space?
Short answer
Find each area: length × width for the rectangle and side × side for the square. Then compare the two numbers.
Know first: Multiplication facts, Square numbers
Step 1 of 5
Area is length times width
Area measures the space inside a flat shape, in square units such as cm². For a rectangle, area = length × width. A square is a rectangle with four equal sides, so its area is side × side, or side².
When a question compares a rectangle with a square, work out both areas. The side lengths alone can mislead you.
| Shape | Formula | Example |
|---|---|---|
| Rectangle | length × width | 3 cm by 12 cm: 36 cm² |
| Square | side × side | side 6 cm: 36 cm² |
A long, thin rectangle and a square can have the same area.
Step 2 of 5
Compute both areas, then compare
Worked example: Rectangle against square
Quantity A is the area of a 5 cm by 9 cm rectangle. Quantity B is the area of a square with side 7 cm. Compare the quantities.
- AQuantity A is greater.
- BQuantity B is greater.
- CThe two quantities are equal.
- DThe relationship cannot be determined.
- 1
Rectangle
5 × 9 = 45 cm².
- 2
Square
7 × 7 = 49 cm².
- 3
Compare
49 is greater than 45, so the square has the greater area.
- 4
See why the other choices are there
A looks greater if you compare the longest sides, 9 and 7. Equal would need both areas to match. Cannot be determined doesn't fit, since every length is given.
Answer
Quantity B is greater.
Wrong: The rectangle has a 9 cm side and the square's side is only 7 cm, so the rectangle is bigger.
Right: The rectangle's area is 45 cm² and the square's is 49 cm², so the square is bigger.
One long side says nothing about area by itself. Multiply both dimensions before comparing.
Step 3 of 5
Multiply before you compare
Check yourself · Question 1
Quantity A is the area of a 2 m by 18 m rectangle. Quantity B is the area of a square with side 6 m. Compare the quantities.
Answer: C
Rectangle: 2 × 18 = 36 m². Square: 6 × 6 = 36 m². The areas are equal.
Trap answer: Quantity A is greater.
Longest side wins
- Why it looks right
- An 18 m side is three times the square's side, so the rectangle looks much larger.
- Why it's wrong
- The rectangle is only 2 m wide. Its area is 2 × 18 = 36 m², exactly the square's area.
- The right answer
- The two quantities are equal.
Check yourself · Question 2
Quantity A is the area of a 9 ft by 11 ft rectangle. Quantity B is the area of a square with side 10 ft. Compare the quantities.
Answer: B
Rectangle: 9 × 11 = 99 ft². Square: 10 × 10 = 100 ft². The square is greater by 1 ft².
Common mistake
"Same perimeter means same area."
- Why it's tempting
- The two shapes use the same total length of edge, so they seem the same size.
- Do this instead
- Compute each area. A square encloses more space than any other rectangle with the same perimeter.
Step 4 of 5
What to remember
Remember
- Rectangle area is length × width; square area is side × side.
- Compare areas, not side lengths.
- Equal perimeters do not mean equal areas.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
What were the two areas in the practice question, and what would you conclude if you only compared side lengths?