Construction of Square

Constructing geometric figures is a key part of mathematics. One of the simplest figures to construct is a square. A square has four equal sides and right angles at each corner. In this article, we will look at the steps to construct a square using different methods.

Introduction

The ability to construct geometric figures is essential for students, as it aids in developing spatial reasoning and visualization skills. Constructing a square is a basic but crucial skill that helps students understand the properties of squares and related geometric shapes.

What is a Square?

A square is a polygon with four equal sides and four 90-degree angles. It is a regular polygon with four lines of symmetry. The diagonal of a square splits it into two congruent right triangles, and its length is the square root of two times the length of a side.

Constructing a Square with a Compass

Creating a square with a compass involves drawing circles and connecting their intersection points. Here are the steps for constructing a square using a compass:

1. Draw a Line Segment: Draw a straight line segment of any length and label it AB.
2. Draw the First Circle: Place the compass point on A and draw a circle with radius AB.
3. Draw the Second Circle: Without changing the compass width, place the compass point on B and draw another circle that intersects the first circle at point C.
4. Draw the Third Circle: Keeping the same compass width, place the compass point on C and draw a circle with radius AC.
5. Draw the Fourth Circle: Again, without changing the compass width, place the compass point on A and draw another circle that intersects the previous circle at point D.
6. Connect the Points: Draw straight lines connecting A to C, C to D, D to B, and B to A.

Constructing a Square in a Circle

Creating a square inside a circle involves drawing the square so that its corners touch the circle. Here’s a simple method to do this:

1. Draw a Circle: Draw a circle of any size and mark its center as point O.

2. Draw the Diameter: Draw a straight line through the center O, making the diameter of the circle. Label the endpoints of the diameter as A and B.

3. Draw a Perpendicular Diameter:

   • Use a compass to draw another circle with the same radius as the first one. Place the pointed end on point A and draw a circle.
   • Without changing the compass width, place the pointed end on point B and draw another circle intersecting the first circle. Label the intersection points as C and D.

4. Draw the Perpendicular Line: Draw a straight line through points C and D. This line should pass through the center O and is perpendicular to AB. Label the intersections of this line with the circle as points E and F.

5. Connect the Points: Draw straight lines connecting points A to E, E to B, B to F, and F to A.

Solved Example

Q. Construct a square with a side length of 5 cm using a compass and straightedge.

Solution:

• Draw a line segment AB using the straightedge.
• With the compass, draw a circle with center at point A and radius of 5 cm.
• Without changing the compass width, draw another circle with center at point B and intersecting the first circle at point C.
• Draw line segments AC and BC using the straightedge.
• Finally, draw line segments BD and AD to complete the square.

Now you have a square ABCD with side length of 5 cm.

Note that step 3 is crucial, as it ensures that points C and D are equidistant from both A and B, thus ensuring that the angles at C and D are both right angles, and that all four sides of the square are of equal length.