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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

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

## FAQs

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

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?

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.

## Recommended Reading

- How to Construct a Rectangle Using Compass and Ruler – Methods, Steps & Examples
- What is a Scalene Triangle – Definition, Properties & Examples
- What is an Isosceles Triangle – Definition, Properties & Examples
- What is an Equilateral Triangle – Definition, Properties & Examples
- What is a Rectangle – Definition, Types, Properties & Examples
- What is a Rhombus – Definition, Types, Properties & Examples
- What is a Trapezium – Definition, Types, Properties & Examples
- What is a Parallelogram – Definition, Properties & Examples
- Types of Quadrilaterals and Their Properties(With Definitions & Examples)
- What is Quadrilateral in Math(Definition, Shape & Examples)
- Angle Bisector of a Triangle – Definition, Properties & Examples
- Similarity of Triangles Criteria – SSS, SAS, AA
- Congruence of Triangles Criteria – SSS, SAS, ASA, RHS
- Altitude of a Triangle(Definition & Properties)
- Median of a Triangle(Definition & Properties)
- How to Construct a Triangle(With Steps, Diagrams & Examples)
- Median of a Triangle(Definition & Properties)
- Altitude of a Triangle(Definition & Properties)
- Congruence of Triangles Criteria – SSS, SAS, ASA, RHS
- Similarity of Triangles Criteria – SSS, SAS, AA
- Angle Bisector of a Triangle – Definition, Properties & Examples
- What is Quadrilateral in Math(Definition, Shape & Examples)
- Properties of Triangle – Theorems & Examples
- Types of Triangles – Definition & Examples
- What is Triangle in Geometry – Definition, Shapes & Examples
- Pair of Angles – Definition, Diagrams, Types, and Examples
- Construction of Angles(Using Protractor & Compass)
- Types of Angles in Maths(Acute, Right, Obtuse, Straight & Reflex)
- What is an Angle in Geometry – Definition, Properties & Measurement
- How to Construct a Tangent to a Circle(With Steps & Pictures)
- Tangent of a Circle – Meaning, Properties, Examples
- Angles in a Circle – Meaning, Properties & Examples
- Chord of a Circle – Definition, Properties & Examples
- How to Draw a Circle(With Steps & Pictures)
- What is a Circle – Parts, Properties & Examples
- How to Construct a Perpendicular Line (With Steps & Examples)
- How to Construct Parallel Lines(With Steps & Examples)
- How To Construct a Line Segment(With Steps & Examples)
- What are Collinear Points in Geometry – Definition, Properties & Examples
- What is a Transversal Line in Geometry – Definition, Properties & Examples
- What are Parallel Lines in Geometry – Definition, Properties & Examples
- What is Concurrent lines in Geometry – Definition, Conditions & Examples
- What is Half Line in Geometry – Definition, Properties & Examples
- What is a Perpendicular Line in Geometry – Definition, Properties & Examples
- Lines in Geometry(Definition, Types & Examples)