HSPT: Comparisons: perimeter of rectangles and squares
Learn to compare the perimeters of a rectangle and a square without mixing perimeter up with area.
How do you compare the distance around a rectangle with the distance around a square?
Short answer
Add all four sides of each shape: 2 × (length + width) for the rectangle and 4 × side for the square. Then compare.
Know first: Adding and multiplying whole numbers, The difference between area and perimeter
Step 1 of 5
Perimeter is the distance around
Perimeter is the total length of a shape's outside edge, measured in plain units such as meters. A rectangle has two lengths and two widths, so its perimeter is 2 × (length + width). A square has four equal sides, so its perimeter is 4 × side.
Perimeter and area are different measurements. A shape can have the bigger perimeter and the smaller area, so always use the formula the question asks for.
| Shape | Formula | Example |
|---|---|---|
| Rectangle | 2 × (length + width) | 8 m by 3 m: 2 × 11 = 22 m |
| Square | 4 × side | side 7 m: 28 m |
Add the length and width first, then double. Every side gets counted.
Step 2 of 5
Count every side, then compare
Worked example: Same edge, different shapes
Quantity A is the perimeter of a 9 m by 5 m rectangle. Quantity B is the perimeter of a square with side 7 m. Compare the quantities.
- AQuantity A is greater.
- BQuantity B is greater.
- CThe two quantities are equal.
- DThe relationship cannot be determined.
- 1
Rectangle
2 × (9 + 5) = 2 × 14 = 28 m.
- 2
Square
4 × 7 = 28 m.
- 3
Compare
28 = 28, so the perimeters are equal.
- 4
See why the other choices are there
B looks greater if you compare areas instead: 45 m² against 49 m². A looks greater if you notice only the 9 m side. Cannot be determined doesn't fit, since every length is given.
Answer
The two quantities are equal.
Find the wrong step
A student compares the perimeter of a 10 m by 3 m rectangle (Quantity A) with the perimeter of a square with side 6 m (Quantity B).
- 1
Adds the length and width, 10 + 3 = 13 m, and calls that the rectangle's perimeter.
What went wrong
10 + 3 covers only two of the four sides. The perimeter is 2 × 13 = 26 m.
- 2
Finds the square's perimeter: 4 × 6 = 24 m.
- 3
Concludes that Quantity B is greater.
The fix
The rectangle's perimeter is 26 m, which beats 24 m. Quantity A is greater.
Step 3 of 5
Use the perimeter formula, not area
Check yourself · Question 1
Quantity A is the perimeter of a 12 m by 2 m rectangle. Quantity B is the perimeter of a square with side 5 m. Compare the quantities.
Answer: A
Rectangle: 2 × (12 + 2) = 28 m. Square: 4 × 5 = 20 m. The rectangle's perimeter is greater.
Trap answer: Quantity B is greater.
Area instead of perimeter
- Why it looks right
- Multiplying the sides is a strong habit, and 5 × 5 = 25 beats 12 × 2 = 24.
- Why it's wrong
- Those are areas. Perimeter adds up the edges: 28 m around the rectangle and 20 m around the square.
- The right answer
- Quantity A is greater.
Check yourself · Question 2
Quantity A is the perimeter of an 11 ft by 5 ft rectangle. Quantity B is the perimeter of a square with side 9 ft. Compare the quantities.
Answer: D
Rectangle: 2 × (11 + 5) = 32 ft. Square: 4 × 9 = 36 ft. The square's perimeter is greater.
Common mistake
"I multiply length by width to compare the shapes."
- Why it's tempting
- Length × width is the first formula most students learn for rectangles.
- Do this instead
- Read the word in the question. Perimeter means add all four sides; area means multiply.
Step 4 of 5
What to remember
Remember
- Rectangle perimeter is 2 × (length + width); square perimeter is 4 × side.
- Count all four sides: length + width is only half the trip.
- Perimeter and area can rank two shapes differently, so use the one asked for.
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 perimeters in the practice question, and would the answer change if you compared areas instead?