the diagonal of a square is 18cm what is the area
MathBot Answer:
The area of a square is $162\, \text{cm}^2$.
Note: Approximate solutions are rounded to the nearest hundredths place.
Given: $$\begin{aligned}d&=18\end{aligned}$$
Area
The area of a square is given by the equation: $$\begin{aligned}A&=s^{2}\end{aligned}$$ where $A$ is area and $s$ is side.
Need:$$\begin{aligned}d&=s \sqrt{2}\end{aligned} \quad \Rightarrow \quad \begin{aligned}18&=s \sqrt{2}\end{aligned} \quad \Rightarrow \quad \begin{aligned}s&=9 \sqrt{2} \approx 12.73\end{aligned}$$
Solution:$$\begin{aligned}A&=s^{2}\\A&=\left(9 \sqrt{2}\right)^{2}\\A&=162\end{aligned}$$
A square is a two dimensional polygon with four equal sides, four right angles, and four vertices. Opposite sides are parallel to each other. Each side has a length of s. Less commonly, one pair of opposite sides is the length l, and the other pair is the width w, or base b and height h, respectively. The diagonal d is the distance between any two non-adjacent vertices, dividing the square into two congruent right triangles.
The diagonal d of a square is the hypotenuse of the right triangle it forms. $$\begin{aligned} d &= s \sqrt{2} = \sqrt{s^{2} + s^{2}} \\ &= \sqrt{l^{2} + w^{2}} \\ &= \sqrt{b^{2} + h^{2}} \end{aligned}$$
The perimeter P of a square is the sum of all the sides. $$\begin{aligned} P &= 4 s \\ &= 2 l + 2 w = 2 (l + w) \\ &= 2 b + 2 h = 2 (b + h) \\ &= 2 \sqrt{2} d \end{aligned}$$
The area A of a square is the region enclosed inside it. $$\begin{aligned} A &= s^{2} \\ &= l w \\ &= b h \\ &= \frac{1}{2} d^{2} \end{aligned}$$
Classification: Polygon, Quadrilateral, Trapezoid, Parallelogram, Rectangle, Rhombus