How to Construct a Square Using Compass and Ruler – Methods, Steps & Examples

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A square is a 2D closed shape with four equal sides and four vertices. The measure of all four angles in a square is $90^{\circ}$ and its opposite sides are parallel to each other.

Let’s understand how to construct a square using a compass and ruler with steps and examples.

How to Construct a Square Using Compass and Ruler?

A square is a quadrilateral with sides of equal length and angles of the same measure of $90^{\circ}$. There are two methods of constructing a square depending on the parameter known for construction. You can construct a square if

• length of a side is known
• length of a diagonal is known

How to Construct a Square With a Given Side?

Since the length of each of the four sides of a square are equal and the measure of each of the four angles is $90^{\circ}$, therefore you can draw a unique square by knowing only the length of one side of a square. 

The following are the steps to construct a unique square with a given side.
Step 1: Draw a line segment $\text{AB}$ of a given length.

Step 2: Extend line $\text{AB} to the right.

Step 3: Place the pointer of the compasses on $\text{B}$ and stretch it to any convenient width. Draw an arc on each side of $\text{B}$, at the points $\text{E}$ and $\text{F}$.

Step 4: Place the compasses at a point $\text{F}$ and with any convenient width, draw an arc above point $\text{B}$.

Step 5: Without changing the compasses’ width, place the compasses on $\text{G}$ and draw an arc above $\text{B}$, crossing the previous arc, and creating point $\text{G}$

Step 6: Join the pints $\text{B}$ through $\text{G}$

Step 7: Set the compasses to $\text{A}$ and set their width to $\text{AB}$. This width will be held unchanged as we create the square’s other three sides

Step 8: Draw an arc above point $\text{A}$.

Step 9: Without changing the width, move the compasses to point $\text{B}$. Draw an arc across $\text{BG}$ creating point $\text{C}$ – a vertex of the square.

Step 10: Without changing the width, move the compasses to $\text{C}$. Draw an arc to the left of $\text{C}$ across the exiting arc, creating point $\text{D}$ – a vertex of the square.

Step 11: Join the points $\text{C}$, $\text{D}$ and $\text{A}$, $\text{D}$

$\text{ABCD}$ is a square where each side has a length of $\text{AB}$.

How to Construct a Square With One Diagonal Given?

Since, the diagonals of a square are equal and are perpendicular to each other, and also bisect each other. Therefore, if you know the length of one of the diagonals of a square, you can construct a unique square.

The following are the steps to construct a square with a given diagonal.

Step 1: Construct a line $\text{AC}$ as a given diagonal of the square

Step 2: Locate $\text{M}$ as a midpoint of the diagonal $\text{AC}$

Step 3: Construct the perpendicular line at $\text{M}$

Step 4: Construct a circle centred at $\text{M}$ and has a radius $\text{MC}$

Step 5: Name the intersection points between the circle and the perpendicular line at $\text{M}$ as $\text{B}$ and $\text{D}$

Step 6: Join the four points $\text{A}$, $\text{B}$, $\text{C}$ and $\text{D}$ to get the required square

Practice Problems

1. What is a square in geometry?
2. Construct a square of side $4 \text{cm}$.
3. Construct a square of diagonal $6 \text{cm}$.

FAQs

Conclusion

A square is a quadrilateral with sides of equal length and angles of the same measure. There are two methods of constructing a square depending on the parameter known for construction. You can construct a square if the length of a side is known or the length of a diagonal is known.

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