Question:easy

If the ratio of the areas of two squares is 25:36, then the ratio of their perimeters is

Show Hint

Area goes with the square of the side, perimeter goes with the side itself. So take the square root of 25:36.
Updated On: Jul 17, 2026
  • 5:6
  • 25:36
  • 6:5
  • 36:25
Show Solution

The Correct Option is A

Solution and Explanation

Step 1: Build actual squares that satisfy the condition.
Instead of algebra, just pick numbers that work. We need two areas in the ratio $25 : 36$. The simplest choice is an area of 25 square units and an area of 36 square units.

Step 2: Get their sides.
For a square, side $=\sqrt{\text{area}}$. So
\[ a = \sqrt{25} = 5, \qquad b = \sqrt{36} = 6 \]
Both areas are perfect squares, which is exactly why the paper chose 25 and 36.

Step 3: Get their perimeters.
Perimeter of a square is four times the side, so
\[ P_1 = 4 \times 5 = 20, \qquad P_2 = 4 \times 6 = 24 \]

Step 4: Reduce the perimeter ratio.
\[ \frac{P_1}{P_2} = \frac{20}{24} \]
Divide both numbers by their highest common factor 4:
\[ \frac{20}{24} = \frac{5}{6} \]
So the perimeters are in the ratio $5:6$.

Step 5: Note the general rule.
For any two similar figures, if the areas are in the ratio $m : n$, the corresponding lengths (side, perimeter, diagonal, radius) are in the ratio $\sqrt{m} : \sqrt{n}$. Turned around, if lengths are in ratio $p : q$, areas are in ratio $p^2 : q^2$ and volumes in ratio $p^3 : q^3$. Knowing this saves the whole calculation.

Step 6: Eliminate the distractors.
$25:36$ is the area ratio itself and would be right only if the question asked for areas again.
$6:5$ and $36:25$ both put the larger square first, contradicting the order set by the question, where the square of area 25 is named first.
Only $5:6$ survives both the square-root step and the order check.

Final Answer:
Since perimeter scales with the side, the ratio is the square root of the area ratio.
\[ \boxed{5:6} \]
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