Imagine you want to move. You want to carry your TV through a window. The window is square and has a side length of \begin{align} \color{#1478c8}a&=\color{#1478c8}70 \text{ cm.}\end{align} Your TV is \(\text{95 cm}\) high and therefore definitely does not fit through the window without tipping it over. But what does it look like when you carry it diagonally through the window?
The length of the diagonals of the window will tell you whether the television will fit through the window. You can find out here how you determine this and what the diagonal of a square is all about in general!
square – basic knowledge
That square is a geometric figure from the quadrilateral family.
That square is defined by the following characteristics:
- All sides are the same length.
- Everyone interior angle is 90 degrees. So the square is equiangular.
The inscription of a square is always the same:
- The pages are labeled with lowercase letters. Since all four sides are the same length, each side is included a labeled.
- The vertices are labeled in capital letters and counterclockwise: A, B, C, D
Figure 2: Definition square
diagonal square – General
In geometry, a diagonal represents a line that connects two opposite corner points of a surface without being a side of the figure.
A square has two diagonals dwhich are each other in Midpoint M of the figure at a right angle and divide the square into two equal right triangles.
The diagonals are labeled d and are often shown as dashed lines.
Figure 3: Diagonals of a square
Diagonal Square – Draw
If you want to draw the diagonals d into a square yourself, connect the opposite corners A and C as well as B and D with each other. The angles between the respective sides of the square and the diagonal d must always correspond to 45 degrees.
Figure 4: Drawing the diagonal of a square
Calculate diagonal square – formula
The diagonal represents the longest line within a square.
You can calculate the length of the diagonal d of a square with side length a using the following formula:
\
You now have all the information to solve the relocation problem from the beginning of the explanation.
How was that again? You want to move and carry a TV through a square window. The window has a side length of \begin{align} \color{#1478c8}a&=\color{#1478c8}70 \text{ cm.}\end{align} Your television is \(\text{95 cm}\) high and therefore definitely does not fit through the window without tipping it over. But does the TV fit through the window at an angle?
So you are looking for the longest distance within the window – the diagonal of a square. If this is longer than the height of the TV, the TV will fit through the window.
To calculate the diagonals, you now use the length of the side of the square for a in the formula. So here \(\text{70 cm}\). You can then solve this calculation.
\begin{align} d &= \sqrt{2} \cdot \color{#1478c8}a\\\\d&= \sqrt{2} \cdot \color{#1478c8}70 \text{ cm} \\ d& \approx 99 \text{ cm} \\ \\ \color{#fa3273}99 \text{ cm} &> \color{#00dcb4}95 \text{ cm} \end{align}
The diagonal d of the window is just \(\text{99 cm}\) long. Accordingly, the television with a height of \(\text{95 cm}\) just fits through the window.
So if you want to calculate the length of the diagonal d of a square, you only need information about the side lengths. But what about the other way around? There are also problems where the diagonal d is given and the side lengths are asked.
Calculate the side length of the square using the diagonal
To calculate the length of one side of a square with a given diagonal, rearrange the above formula for the length of the diagonal d according to the side length a.
\begin{align}d&=\sqrt{2}\cdot a &&|: \sqrt{2}\\ \frac{d}{\sqrt{2}}&=a \end{align}
You can calculate the length of the sides a of a square with the diagonal d using the following formula:
\
You can decide for yourself whether you use the formula directly to calculate the side lengths, or change the formula for the length of the diagonals if necessary.
Task 1
Imagine you want to buy a new bag for your laptop. Only the face diagonal of the bag is included on the website \begin{align}d= 60 \text{ cm} \end{align} specified. The longest side of your laptop is \(\text{37 cm}\). Would the laptop fit in the bag? The bag can be considered square.
solution
So we are looking for the side length of the square pocket. For the calculation you use the formula for the length of the diagonal \(d=60\,cm\) and solves for side a.
\begin{align} {\color{#fa3273}d} &= \sqrt{2} \cdot a \\ \\ {\color{#fa3273}60\text{ }cm} &= \sqrt{2} \ cdot a && |: \sqrt{2} \\42.43 \text{ } cm & \approx a \\ \\42.43 \, cm &> 37\, cm\end{align}
With a side length of over \(\text{42 cm}\) the bag is big enough for the laptop. There might even be some space left for the charging cable.
By changing the formula for the length of the diagonal to the side length a, you can now also calculate the perimeter U or area A of a square if you only know the length of the diagonal.
Calculate the area of a square with diagonals
The formula for calculating the area of a square depends on the side length a.
The area A of a square with side length a can be calculated using the following formula:
\begin{align}A&=a \cdot a \\ &=a^2\end{align}
If you only gave the diagonal d and you want to determine the area of the square from it, you must first determine the side length a as in the previous example. The calculation consists of two steps.
- Step 1: Rearrange the formula for the length of the diagonal d according to the side length a \
- step 2: Calculate the area A with the determined side length a \
These two steps can also be combined in a formula for the area.
The area A of a square with the length of the diagonal d can be calculated using the following formula: \begin{align}A&=\frac{d}{\sqrt{2}} \cdot \frac{d}{\sqrt{2} }\\ &=\left(\frac{d}{\sqrt{2}}\right)^2\end{align}
So you can either remember the formula for the area of a square as a function of the diagonal or use the two steps above.
exercise 2
Imagine you are currently in a furniture store to buy new curtains for your square window. Unfortunately, you forgot the dimensions of the window. You just remember the approximate length of the diagonals with \begin{align}{\color{#fa3273}d} \approx {\color{#fa3273}99 \text{ cm}} ,\end{align} you used when move for the television have already calculated.
How much area of fabric for the curtains do you need at least so that the window is completely covered?
solution
Wanted is the Area A of the window.
Step 1: Calculate side length a
Write the formula for the diagonal d according to the side length a um.
\begin{align} {\color{#fa3273}d} &= \sqrt{2} \cdot a \\ \\ {\color{#fa3273}99\text{ }cm} &= \sqrt{2} \ cdot a && |: \sqrt{2} \\60 \text{ } cm & \approx a \end{align}
Step 2: Calculate the area A
Set the side length determined in step 1 \into the formula for the area.
\begin{align} A &= {\color{#1478c8}a}\cdot {\color{#1478c8}a}\\ \\ A &={\color{#1478c8}60\, cm} \cdot { \color{#1478c8}60 \,cm}\\A & =8100 \,cm^2\end{align}
The area of the window is roughly \(\text{8100 } cm^2\) so you need at least this area of fabric for the curtains to completely cover the window.
You can proceed in a similar way if you want to calculate the perimeter \(\bf{U=4\cdot a}\) of a square given the diagonal d.
As you can see, you can use the formula for calculating the length of the diagonals of a square in many ways. But why does this formula apply at all?
diagonal square – formula proof
The formula for the length of the diagonals can be derived using the Pythagorean theorem.
In memory of: The Pythagorean theorem describes a relationship between the three sides a, b and c of a right triangle. The following applies: \(\bf{a^2+b^2=c^2}\), where side c represents the hypotenuse.
The two diagonals of the square each divide the quadrilateral into two right-angled triangles.
Figure 7: Calculation of the length of the diagonal d using the Pythagorean theorem
For the derivation of the formula consider the lower triangle \(\Delta ABC\). the diagonal d is opposite the right angle and thus represents the hypotenuse within the triangle. The remaining sides of the triangle have a length of a.
With this information, you can apply the Pythagorean theorem to the triangle.
\begin{align}{\color{#fa3273}d}^2={\color{#1478c8}a}^2+{\color{#1478c8}a}^2 \end{align}
Now you solve the equation for the diagonal d and simplify it as much as possible.
\begin{align}{\color{#fa3273}d}^2&={\color{#1478c8}a}^2+{\color{#1478c8}a}^2 &&|\sqrt{\text{ }} \\ \color{#fa3273}d&=\sqrt{{\color{#1478c8}a}^2+{\color{#1478c8}a}^2}\\\color{#fa3273}d&=\sqrt{ 2\cdot {\color{#1478c8}a}^2} \\\color{#fa3273}d&=\sqrt{2}\cdot \color{#1478c8}a \end{align}
You have successfully derived the formula for the length of the diagonals of a square using the Pythagorean theorem.
diagonal square – tasks
Now it’s time to put your knowledge to the test or to deepen it with some tasks on the subject of the diagonal of a square!
task 3
A square has side lengths of \Calculate the length of the diagonal d of the square.
solution
For the calculation you put the side length of the square into the formula for the length of the diagonal d.
\begin{align} d &= \sqrt{2} \cdot \color{#1478c8}a\\\\d&= \sqrt{2} \cdot \color{#1478c8}40 \text{ cm} \\ d& \approx \text{56.57 cm} \end{align}
The length of the diagonal of the square is approximately \(\text{56.57 cm}\).
task 4
The area of a square is \What is the length of the diagonal d of the square?
solution
Put the area in the formula for the area of a square depending on the diagonal d \(A=1600\,cm^2\) and solve for the diagonal d.
\begin{align}\color{#1478c8} A&=\left(\frac{d}{\sqrt{2}}\right)^2 \\\\ {\color{#1478c8}2500 \, cm^2 } &=\left(\frac{d}{\sqrt{2}}\right)^2&&|\sqrt{\text{ }} \\50 \, cm&=\frac{d}{\sqrt{2} }&&|\cdot \sqrt{2}\\50 \, cm\cdot \sqrt{2} &=d\\\text{70.71 cm} &\approx d\end{align}
The diagonal d thus has a length of about \(\text{70.71 cm}\).
square diagonal – Key facts at a glance
- In geometry, a diagonal \(d\) represents a line that connects two opposite corner points of a surface without being a side of the figure
- A square has two diagonals d, which intersect at right angles at the center M of the figure, each dividing the square into two equal right triangles.
- You can calculate the length of the diagonal d of a square with side length a using the following formula:
\
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To measure the length of one side of a square with…