

Introduction to Construction of a Square
In Geometry, a square is a two-dimensional figure with four equal sides and four equal angles. Each angle of the square measures 90 degrees. Also, the diagonals of a square are equal and bisect each other at 90 degrees. The square is considered a special type of rectangle as all the properties of the square are quite similar to the properties of a rectangle. The only difference between the two is that the rectangle has only its opposite sides equal. Hence, the rectangle will be called a square if the length of all its four sides is equal.
Read the article below to learn the steps of construction of a square with a compass and a ruler and steps of construction of a square inscribed in a circle.
Construction of a Square with Compass and Ruler
A square is a closed, two-dimensional figure with four equal sides. The four angles of a square are also equal i.e. 90 degrees each. The only dimension we have been provided is the length of one side of a square i.e. 5 cm. We know that the interior angles of a square are at right angles or equal to 90 degrees. Hence, we do not require any other measurement for constructing a square. The measurement of the length of all the four sides of a square equal and perpendicular to each other. With the help of this information, we can now construct a square each of whose sides measure 5 cm.
Construction of a Square with a Given Side of 5 cm
Below are the steps for constructing a square with a compass whose each side measure 5 cm:
Draw a line segment AB of 5 cm with the help of the ruler.
Extend the line segment AB. Place the pointer of the compass on point B, and with convenient width, draw two arcs on each side of B as shown in the figure given below. Mark the point as F and G.
With F as the center and the radius 5 cm draw an arc above point B.
With G as a center and the same radius of the compass, draw on the arc above point B, crossing the previous arc. Mark the point H, where both the arcs meet as shown in the figure below.
Join point B and H
Taking A as the center and radius 5 cm, draw an arc above the point A.
Now with the same radius and taking B as center, draw an arc across BH as shown in the figure below, and mark the point as C. This is the vertex of a square.
Now with the same radius and taking C as the center, draw an arc to the left of C, crossing the previous arc. Mark the point D where both the arc meets.
Join the point CD and AD to get the desired square ABCD of each side 5 cm.
Square Inscribed in a Circle
A square is drawn inside a circle or inscribed in a circle if its four vertices lie on the circumference of a circle. The diagonals of a square are always equal to the diameter of a circle. The figure given below shows the square inscribed in a circle.
Construction of a Square Inscribed in a Circle
Following are the steps to construct a square inscribed in a circle
Draw a circle with center O with the help of the compass.
Mark a point A on the edge of the circle as shown in the figure given below. This will be considered as one of the vertices of a square.
Using the ruler, draw the diameter of a circle from point A that passes through the center of the circle, creating a new ending point C.
Taking A as center and radius more than half of the length AC, draw an arc both above and below the point O.
Taking C as center and with the same radius, draw an arc both above and below the point O that will intersect the arc drawn in step 5.
Draw a line through the point where opposite arcs meet each other. Extends the line long enough so that it touches the circle both at the top and bottom creating the new points B and D. The line segment BD will also be the diameter of the circle.
Join each successive point i.e. A, B, C, and D to form a square as shown in the figure below.
Hence, you will get a square inscribed in the given circle.
Solved Examples
1. A square with side length a is inscribed in a circle. Determine the radius of a circle in terms of the side length of square a.
Solution:
The diagonals of a square are the diameter of a circle. Let d and r be the diameter and radius of a circle respectively. According to the Pythagorean theorem, we have
d² = a² + a²
d² = 2a²
D = \[\sqrt{2a^{2}}\]
= \[a\sqrt{2}\]
As we know, the diameter of a circle is twice its radius. So
R = d/2
= \[a\sqrt{2}/2\]
2. Construct a square ABCD whose measurement of the length of the diagonals is 6 cm. Measure the sides of a square.
Solution:
As we know, all four sides of a square are equal, and also each angle of the square measures 90 degrees each. We also know that the diagonals of a square are equal and perpendicularly bisect to each other.
With these properties, we can construct a square whose diagonal is 6 cm in length. Following are the steps to construct a square whose diagonal is 6 cm.
Draw a line segment AC of 6 cm with the help of the ruler.
Construct a perpendicular bisector XY through the midpoint of AB.
The perpendicular bisector XY will intersect the line segment AC at O. We get OC = OA= 3 cm.
Taking O as center and radius 3.cm, draw two arcs cutting the line XY at points B and D.
Join the line segments AB, BC, CD, and DA. Hence, ABCD is the desired square of diagonal 6 cm as shown below.
Measuring the sides of a square using a ruler, we get AB = BC = CD = DA = 4.2 cm.
FAQs on Construction of a Square
1. What are the key properties of a square that are essential for its construction?
For a precise geometrical construction, you must use the fundamental properties of a square. The most important ones are:
All four sides are of equal length.
All four interior angles are perfect right angles (90 degrees).
The two diagonals are equal in length.
The diagonals bisect each other at a 90-degree angle.
2. What are the basic tools required for the geometrical construction of a square?
To construct a square accurately using geometrical principles, you will need the following tools:
A ruler (or a straightedge): For drawing straight line segments of specific lengths.
A compass: For drawing arcs, circles, and marking off equal lengths.
A protractor: To measure and draw angles, specifically the 90° angles at the vertices.
A sharp pencil: For making clear and precise markings.
3. How do you construct a square when only the length of one side is given?
When given a side length, say 's', you can construct a square by following these steps:
Draw a line segment AB of length 's'.
At point A, construct a perpendicular line (a 90° angle) to AB.
Using your compass, set the width to 's' and draw an arc from point A along the perpendicular line. Mark this intersection as point D.
Now, with the same compass width 's', draw an arc from point B and another from point D. The point where these two arcs intersect is point C.
Join points B to C and D to C to complete the square ABCD.
4. How is the construction of a square different if you are given the length of its diagonal instead of a side?
If you are given the diagonal length 'd', the construction method changes significantly because it relies on the properties of diagonals:
Draw the diagonal AC of length 'd'.
Construct the perpendicular bisector of AC. This line will pass through the midpoint of AC, let's call it O.
The other diagonal, BD, also has length 'd' and passes through O. So, set your compass to a radius of d/2 (half the diagonal length).
With the compass point at O, draw arcs on the perpendicular bisector on both sides of AC. Mark these points as B and D.
Join the points A, B, C, and D to form the required square.
5. Why is constructing a 90-degree angle a mandatory step when building a square from a given side?
Constructing a 90-degree angle is mandatory because it is a defining property of a square. A quadrilateral with four equal sides but without 90-degree angles is a rhombus, not a square. The right angle ensures that adjacent sides are perpendicular, which in turn guarantees that opposite sides are parallel and all angles are equal, fulfilling the complete definition of a square.
6. How does the construction of a square differ from the construction of a rhombus?
The key difference lies in the angles. Here's a comparison:
Square Construction: Requires constructing four 90-degree angles. Its diagonals are equal and bisect each other at 90 degrees.
Rhombus Construction: Does not require 90-degree interior angles. You can construct it if you know a side length and one interior angle (which isn't 90°), or if you know the lengths of its two unequal diagonals, which still bisect each other at 90 degrees.
Essentially, a square is a specific type of rhombus where all angles are 90 degrees.
7. What is the importance of diagonals when constructing a square inscribed in a circle?
The diagonals are critically important because they act as a bridge between the properties of the square and the circle. For a square inscribed in a circle:
The two diagonals of the square are equal to the diameter of the circle.
They intersect at the centre of the circle at a 90-degree angle.
This simplifies the construction greatly. You can simply draw two perpendicular diameters in the circle and connect their endpoints on the circumference to form a perfect square.

















