Diagonal Length Calculation for Square Area
The problem asks us to find the length of the diagonal of a square given its area. We are given the area of the square is \(32\) cm\(^2\).
Finding the Side Length
The formula for the area (\(A\)) of a square with side length (\(s\)) is \(A = s^2\). We can use the given area to find the side length:
\(s^2 = 32 \, \text{cm}^2\)
To find \(s\), we take the square root of both sides:
\(s = \sqrt{32} \, \text{cm}\)
Simplify the square root:
\(s = \sqrt{16 \times 2} \, \text{cm} = 4\sqrt{2} \, \text{cm}\)
Calculating the Diagonal Length
The relationship between the side length (\(s\)) and the diagonal (\(d\)) of a square is given by the Pythagorean theorem, which simplifies to \(d = s\sqrt{2}\).
Now, substitute the calculated side length (\(s = 4\sqrt{2}\) cm) into the diagonal formula:
\(d = (4\sqrt{2}) \times \sqrt{2} \, \text{cm}\)
\(d = 4 \times (\sqrt{2} \times \sqrt{2}) \, \text{cm}\)
\(d = 4 \times 2 \, \text{cm}\)
\(d = 8 \, \text{cm}\)
Therefore, the length of the diagonal of the square is \(8\) cm.
Similar Questions
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Important Questions from 2-D Mensuration
- The length of a rectangular plot is $(x^2 + xy + y^2)$ m and its breadth is $(x^2 - 5xy - y^2)$ m.
Find its perimeter, when $x = 1$ and $y = -1$. - The ratio between the perimeter and breadth of a rectangle is 3: 1. If the area of the rectangle is $98 \text{ cm}^2$, find the perimeter (in cm) of the rectangle.
The shorter side of a rectangle is 15 cm less than the longer side. The numerical value of its area is equal to 5 times the numerical value of its perimeter. What is the length (in cm) of its longer side?
- In a circle of radius 10.5 cm, if the angle of a sector is $\frac{2\pi}{3}$, then the perimeter of the sector is (in cm):
(Take $\pi = \frac{22}{7}$) - Find the circumference (in m) of the largest circle that can be inscribed in a rectangle whose dimensions are given as 114 m and 63 m.
Take $\pi = \frac{22}{7}$
