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

how to construct a square

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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.

how to construct a square

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.

how to construct a square

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

how to construct a square

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 $\text{F}$.

how to construct a square

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

how to construct a square

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 $\text{G}$

how to construct a square

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

how to construct a square

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 $\text{A}$.

how to construct a square

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.

how to construct a 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.

how to construct a square

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

how to construct a square

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

girl-with-teacher-happy
Maths can be really interesting for kids

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

how to construct a square

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

CodingHero - How to Construct a Square Using Compass and Ruler - Methods, Steps & Examples Construction of Square 13

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

how to construct a square

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

how to construct a square

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

how to construct a square

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

how to construct a square
Famous Mathematicians

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

Can we construct a square if we know the length of its diagonal?

how to construct a square

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

What is the first step in constructing a square?

how to construct a square

The construction of a square starts with a given line segment $\text{AB}$. It then erects a perpendicular at one end of the line, which will become the second side of the square. The compass is then set to the length of the given side, and the other three sides are marked off.

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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