CBSE Class 7 Maths Chapter 6 Constructions and Tilings Notes 2025-26

In this chapter, you will learn various methods of geometric construction, focusing on techniques like drawing perpendicular bisectors, bisecting angles, and creating patterns and tilings. All of these are done using basic tools like the ruler, compass, or sometimes even a rope, as mentioned in ancient Indian methods. Through these techniques, students build a strong foundation in geometry and spatial reasoning, which are important skills for higher classes.

The chapter starts with the ‘Eyes’ construction, where students create eye-shaped patterns using two arcs. To make the eyes perfectly symmetrical, both arcs must be drawn using centers A and B such that distances from each center to key points are equal (AX = AY = BX = BY). The line joining A and B acts as the perpendicular bisector of the base line XY, passing through its midpoint, and ensuring symmetry in the drawing. This visual and concrete activity helps to understand the concepts of symmetry and bisectors.

Bisection means dividing any line segment or angle into two equal parts. A perpendicular bisector is a special line that cuts a given line into two identical parts at 90 degrees. The construction of a perpendicular bisector without measuring—by just using a compass and an unmarked ruler—shows that geometric precision is possible using basic tools.

To draw the perpendicular bisector of a line segment XY, use a compass to draw arcs of equal radii from points X and Y above and below the line. The intersecting points mark positions A and B. Connecting these creates a line AB, which is the required perpendicular bisector. This line passes through the midpoint of XY and stands at right angles to it. With this method, accuracy is maintained without depending on measurements from scales.

• A point lying at the same distance from X and Y will always be somewhere on the perpendicular bisector of XY.
• Various designs and patterns can be created by changing the points from which arcs are drawn but keeping them on the perpendicular bisector for symmetry.

Constructing a right angle at a given point of a line uses the perpendicular bisector method too. By marking points X and Y at equal distances from O (the given point on the line), and finding their perpendicular bisector, a perfect 90° angle is achieved at O. The chapter also introduces the Śulba-Sūtras, ancient Indian mathematical texts, which describe how a rope can be used for the same purpose on the ground—showing geometric construction is timeless and universal.

An angle bisector divides an angle into two equal smaller angles. Given an angle XOY, equal distances are marked on its arms from O, then arcs of the same radius are drawn from these points to intersect. Joining O to this intersection gives the angle bisector. This can be used to construct multiple patterns and to bisect any given angle further (for example, getting 45° from 90°). The principle uses triangle congruence, ensuring every step is mathematically perfect.

To copy an angle, draw an arc from the vertex of the angle you want to copy, measure the length between its intersection points, and reproduce this at a new location with a compass. This approach ensures the two angles are exactly the same using basic tools—a skill needed for geometric designs and constructions.

Drawing a line parallel to a given line through a particular point is based on constructing equal corresponding angles. To do this, the angle made by the transversal and the original line is replicated at the new point using a compass and ruler. Joining the required point through an equal angle gives the desired parallel line. This geometric technique is widely used in design, tilings, and architectural drawings.

Geometric construction is not just theoretical—it has practical and artistic applications. The chapter describes how to construct trefoil arches and pointed arches, similar to those seen in famous buildings like the Red Fort. These designs use symmetry, equal angles, and arcs with equal radii, and students are encouraged to experiment with variations in these constructions.

The chapter guides students to create regular hexagons, which can be made using six equilateral triangles each with a 60° angle. With a compass and ruler, these shapes can be drawn with high precision. Related shapes, like 6-pointed stars, can also be built step by step, reinforcing geometric principles like symmetry, equal sides, and equal angles.

Tiling is a way of covering a surface completely with a shape or set of shapes, without leaving gaps or overlaps. This concept is applied through puzzles such as tangrams—seven fixed shapes that can be rearranged to make various objects, animals, or symbols. While fun, this also develops an understanding of area, congruence, and spatial reasoning.

The chapter covers tiling rectangles with dominoes, such as covering a 4×6 grid with 2×1 tiles. It explains that a region can be tiled only if the total number of squares is even, and that each domino covers one black and one white square if the grid is colored like a chessboard. If the numbers of black and white squares differ, tiling becomes impossible. This is a practical introduction to mathematical reasoning around parity (even/odd numbers) and area.

• Shapes like squares, triangles, and regular hexagons can tile an entire plane, used in art, nature (like bee hives), and architecture.
• Some tiling patterns are found in daily life, and designs can be created using a combination of geometric shapes.

• Bisection cuts a line or angle into two equal parts; a perpendicular bisector is always at 90° to the segment it divides.
• Any point equidistant from two endpoints of a segment lies on its perpendicular bisector.
• A 90° angle at any point of a line can be constructed using the perpendicular bisector method.
• Angles can be bisected or copied using compass and ruler, often by using congruence properties of triangles.
• Equilateral triangles are used to construct regular hexagons and other polygons.
• Tiling involves covering a region with shapes without gaps or overlap; tiling is possible only under certain conditions linked to shape and parity.

Throughout the chapter, 'Figure it Out' prompts encourage students to actively experiment with construction, understand why geometric rules work, and create their own artistic patterns, making geometry enjoyable and practical.

Class 7 Maths Chapter 6 Notes – Constructions and Tilings: Key Concepts Explained

These Class 7 Maths Chapter 6 notes on Constructions and Tilings are structured for easy revision and step-wise understanding. With clear explanations of perpendicular bisectors, angle bisectors, and tiling patterns, students can quickly revise all essential concepts. Simplified methods and visual examples help strengthen foundational geometry.

By following these comprehensive revision notes, students can master key construction techniques for lines, angles, and symmetry. The material covers practical applications, making math engaging through real-life patterns and puzzles. It's designed to boost speed, accuracy, and confidence for exams and classroom assignments.