What happens to the perimeter of a square if we double its side?
Answer: The perimeter doubles too. Side a gives 4a; side 2a gives 8a = 2 × 4a. Perimeter : side stays 4 : 1 for every square.
Step-by-step solution
Idea: The perimeter of a square is 4 × side, so it changes in the same proportion as the side.
- Side a: perimeter = 4a. Side 2a: perimeter = 4 × 2a = 8a = 2 × (4a). So the perimeter doubles.1 mark
- Perimeter ÷ side = 4a ÷ a = 8a ÷ 2a = 4. The ratio perimeter : side is 4 : 1 for every square, big or small.1 mark
If the side of a square is doubled, its perimeter doubles; the ratio perimeter : side is always 4 : 1.
Check: Side 2 units → perimeter 8; side 4 units → perimeter 16 = 2 × 8 ✓.
Answer to write in the exam
Side a: perimeter = 4a
Side 2a: perimeter = 4 × 2a = 8a = 2 × 4a
∴ The perimeter doubles; perimeter : side = 4 : 1.
Common mistakes that cost marks
- Saying the perimeter becomes four times. That happens to the area, not the perimeter.
- Adding 2 to the perimeter instead of multiplying by 2.
How this can come in the exam
MCQ (1 mark)
If the side of an equilateral triangle is tripled, its perimeter becomes
- the same
- 3 times
- 6 times
- 9 times
Show answer
(B) 3 times
Perimeter = 3 × side, so tripling the side triples the perimeter.
Try one yourself
A square has side 2.5 cm. Find its perimeter, and the perimeter when the side is doubled.
Show answer
4 × 2.5 = 10 cm; 4 × 5 = 20 cm, which is double.
More questions like this
- What about a circle? What is its perimeter (usually called the circumference) in terms of its diameter? Is the ratio of circumference (C) to diameter (D) the same for circles of all sizes (see the figure)? What do you think?
- What is the value of the C/D ratio? How would you estimate this ratio?
- You can do a simple measurement at home to estimate the C/D ratio. Take a cotton reel with thin thread around it. Measure the diameter D of the reel as accurately as possible. Unwrap and then tightly wrap the thread around the reel 20 times. Unwrap it again; measure its length L, and calculate L20D. This is the ratio we want. For accuracy, the thread should be very thin. Please do the experiment! Do you get a ratio between 3 and 4? Between 3.1 and 3.2? It is also possible to estimate the C/D ratio using pure geometry, i.e., without any measurements at all! Can you imagine how?
- The Mesopotamian Hexagon-to-Circle comparison (see the figure). Can you see why this shows that π > 3?
- Archimedes’ method utilising inscribed and circumscribed polygons (see the figure). Can you see why this diagram of an inscribed and circumscribed hexagon tells us that π is between 3 and 2√3? (Hint: Use the Baudhāyana–Pythagoras Theorem.)