How to Draw Perfect Squares and Circles with Just a Ruler and Compass

To construct circles and squares accurately, you use a ruler and compass for circles and a ruler with a set square or protractor for squares. In CBSE Class 6 Mathematics Chapter 8, you learn to draw circles, arcs, squares, and rectangles, and check their properties using basic tools. These constructions help you reproduce curved designs and understand geometric shapes better. By the end of this article, you will know how to use these tools for precise constructions and verify important properties like the equality and bisection of diagonals.

You can draw a circle of a given radius by placing the compass point at the center and adjusting the pencil end to the required radius using a ruler, then rotating the compass to complete the circle. To draw an arc, you use the same method but only draw a part of the circle. For reproducing curved artwork like waves, rings, or eyes, you combine arcs and circles of different sizes and positions. For example, to draw a ring, you first draw a large circle, then a smaller circle with the same center, creating the ring-shaped region between them. In real life, this technique helps you create patterns for rangoli or decorative borders. Many students mistakenly use only freehand drawing, which does not ensure accuracy; using a compass and ruler gives precise results.

To construct a square or rectangle with all right angles, you start by drawing one side with a ruler. Then, use a set square or protractor to make a 90° angle at each corner. For a square, all sides must be equal; for a rectangle, only opposite sides are equal. Continue drawing sides, making sure each angle is a right angle. For example, to draw a rectangle 5 cm by 3 cm, draw a 5 cm line, use a set square to mark a 90° angle at each end, measure 3 cm for the sides, and connect the ends to complete the rectangle. In real life, this method is used to draw the outline of a book or a photo frame. Some students forget to check all angles, leading to a shape that is not a true rectangle or square.

To prove that the diagonals of a rectangle are equal and bisect each other, first construct the rectangle using a ruler and set square. Draw both diagonals from opposite corners. Measure their lengths with a ruler; you will find both diagonals are equal. Next, check where the diagonals cross. Measure the segments from the intersection point to each corner; both halves of each diagonal are equal, showing they bisect each other. For a square, the diagonals are not only equal and bisect each other, but also meet at right angles (90°). For example, if you draw a square of side 4 cm, each diagonal will measure about 5.7 cm, and the intersection divides each diagonal into two equal parts of about 2.85 cm. Many students think only the sides are equal, but the diagonals have these special properties too.

To construct a rectangle or square from a given diagonal, first draw the diagonal using a ruler. For a square, use the property that both diagonals are equal and bisect each other at right angles. Use a compass to draw arcs from each end of the diagonal, with radius equal to half the diagonal, to locate the other two corners. For a rectangle, use the fact that the diagonals are equal and bisect each other, but do not meet at right angles unless it is a square. After constructing the rectangle or square, you can use these shapes to build composite figures, such as a house shape made from a rectangle (for the walls) and a triangle (for the roof). For example, if the diagonal of a square is 7 cm, you can use the formula $\text{side}=\frac{\text{diagonal}}{\sqrt{2}}$ to find the side, then construct the square. Some students forget to check the angles when constructing from the diagonal, which can lead to an incorrect figure.