To find the diagonal of a square using a calculator, the most common approach involves knowing the length of the square's side and applying a simple formula based on the Pythagorean theorem. The core formula relating the diagonal (d) and the side (s) of a square is d = s√2.
Understanding the Diagonal Formula
A square's diagonal divides it into two congruent right-angled triangles. The sides of the square form the legs of these triangles, and the diagonal is the hypotenuse. According to the Pythagorean theorem ($a^2 + b^2 = c^2$), where 'a' and 'b' are the legs and 'c' is the hypotenuse:
$s^2 + s^2 = d^2$
$2s^2 = d^2$
Taking the square root of both sides:
$\sqrt{2s^2} = \sqrt{d^2}$
$s\sqrt{2} = d$
Thus, the length of the diagonal is the side length multiplied by the square root of 2.
Finding the Diagonal from Side Length using a Calculator
If you already know the length of the side (s) of the square, you can easily find the diagonal (d) with a calculator:
- Enter the side length (s) into your calculator.
- Multiply by the square root of 2 ($\sqrt{2}$). Most calculators have a dedicated $\sqrt{x}$ key. You'll typically press the $\sqrt{}$ key, then '2', and then '=' or the multiplication key.
- The result is the length of the diagonal.
Example:
If the side length of a square is 5 units:
- Enter
5 - Press
*(multiply) - Press
√(square root) - Press
2 - Press
=
You will get a result approximately equal to 7.071. So, the diagonal is approximately 7.071 units.
Starting from the Perimeter: As per the Reference
The provided reference outlines how to find the diagonal if you are given the perimeter first.
- You know that the formula of the perimeter is 4s (Perimeter of Square = 4 × side).
- Using this formula, you can find the length of a side (s) by dividing the perimeter by 4: s = Perimeter / 4.
- Then, apply the formula d = s√2, and you will get the length of the diagonal of a square.
This two-step process allows you to arrive at the diagonal length even when only the perimeter is known.
Example using Perimeter:
Suppose the perimeter of a square is 24 units.
- Find the side length:
- s = Perimeter / 4
- s = 24 / 4 = 6 units
- Find the diagonal using the side length:
- d = s√2
- d = 6√2
- Use your calculator:
- Enter
6 - Press
*(multiply) - Press
√(square root) - Press
2 - Press
=
- Enter
You will get a result approximately equal to 8.485. So, the diagonal is approximately 8.485 units.
Formulas At a Glance
Here are the key formulas used:
| Concept | Formula | What it calculates |
|---|---|---|
| Perimeter of Square | $P = 4s$ | Total length of the sides |
| Side from Perimeter | $s = P / 4$ | Length of one side |
| Diagonal of Square | $d = s\sqrt{2}$ or $d = s \times \text{sqrt}(2)$ | Length of the diagonal |
Quick Example Calculations
This table illustrates how to find the side and then the diagonal starting from the perimeter, using the formulas and steps described above:
| Perimeter (P) | Side (s = P/4) | Diagonal (d = s√2) (approx.) |
|---|---|---|
| 12 units | 3 units | $3\sqrt{2} \approx 4.243$ units |
| 20 units | 5 units | $5\sqrt{2} \approx 7.071$ units |
| 36 units | 9 units | $9\sqrt{2} \approx 12.728$ units |
Using Your Calculator Effectively
When calculating $s\sqrt{2}$, some calculators allow you to enter the full expression directly, like 5 * √2 =. Others might require you to calculate $\sqrt{2}$ first (which is approximately 1.414), and then multiply that result by the side length (e.g., 5 * 1.414 =). For the most accurate result, use the $\sqrt{2}$ function directly on your calculator rather than a rounded decimal approximation.
