Constructions and Tilings
Use a ruler and compass to build precise geometric figures — then discover how shapes cover regions without gaps or overlaps.
In this chapter, you will construct perpendicular bisectors, angles, parallel lines, arches and hexagons, then explore tangrams, grid tilings, symmetry and patterns that fill the plane.
By the End of This Chapter, You Will Be Able To:
The Classical Construction Toolkit
In classical Euclidean geometry and the NCERT curriculum, constructions are executed using only two fundamental instruments without measuring numbers:
Draws straight line segments of arbitrary length connecting two given points.
Draws circles, circular arcs, and transfers precise lengths without numerical measurement.
Marks distinct points of intersection between lines, rays, and arcs.
6.1 Geometric Constructions
Eyes, Perpendicular Bisectors, Congruence Proof, Locus, 90°, Śulba-Sūtras, Angles, Arches & Hexagons
The ‘Eyes’ Construction (Textbook Pages 136–137)
How circular arcs drawn from centers A and B produce a symmetrical eye shape
For the upper and lower arcs to match symmetrically along line XY, centres A and B must satisfy:
Therefore, A is equidistant from X and Y, and B is equidistant from X and Y. Connecting A and B produces the perpendicular bisector of XY!
Perpendicular Bisector Construction & Mathematical Proof
Follow the 3-step classical construction and inspect why ΔAOX ≅ ΔAOY
Current Step 3: Draw the straight line AB. Line AB passes through the midpoint O of XY and forms a 90° right angle with XY, bisecting it perpendicularly.
The Equal-Distance Locus Principle
Drag point P horizontally and vertically. Discover that PX = PY happens if and only if P lies exactly on the perpendicular bisector line (x = 200)!
Construction of a 90° Angle at a Given Point (Page 141)
Given point O on a line: mark X and Y equidistant from O ($OX = OY$). Then O is the midpoint of segment XY. Constructing arcs above O gives point A, producing ray OA $\perp$ line with only one pair of arcs!
Ancient Construction Methods — The Śulba-Sūtras
Ancient Indian geometric texts of the Vedic period, known as Śulba-Sūtras ('rules of the cord'), were used for constructing sacrificial fire altars with exact geometric proportions. Instead of a modern compass, ancient geometers used a rope (rajju) fastened to pegs (śanku) at endpoints X and Y. Folding the rope in half marked its midpoint. Stretching the midpoint taut above XY gave point A, and stretching it below gave point B. Line AB is the exact perpendicular bisector!
Angle Bisection Lab & 45° Angle (Pages 143–145)
Cut arcs OA = OB on arms, then intersect arcs from A and B at C
Gap Angle Around a Point (Textbook Page 152)
Angles meeting at point O: 40°, 60°, 50°, 30°, 40°, and 90°. What is the remaining gap angle, and will a 70° angle fit?
Constructing 60° Angles & Regular Hexagons
Six equilateral triangles fitting around a central point ($6 \times 60^\circ = 360^\circ$)
- Draw segment AX.
- With centre A and radius r, draw an arc cutting AX at B.
- With centre B and the exact same radius r, cut the first arc at C.
- Join AC. Since $AB = BC = AC = r$, $\Delta ABC$ is equilateral!
- Therefore, $\angle CAX = 60^\circ$.
6.2 Tiling
Tangrams, 2×1 Domino Tilings, Area Parity, Checkerboard Impossibility Proofs & Plane Tessellations
What is Tiling? (No Gaps, No Overlaps)
Tiling (or tessellation) means covering a region or the flat plane using a set of shapes such that there are zero gaps (no uncovered holes) and zero overlaps (no shapes on top of each other).
Shapes meet edge-to-edge seamlessly.
Uncovered empty space left behind.
Shapes crossing over each other's boundaries.
Chinese Tangram Lab (7 Geometric Pieces)
- 2 Large Triangles (Pieces A & B)
- 1 Medium Triangle (Piece D)
- 2 Small Triangles (Pieces C & E)
- 1 Square (Piece G)
- 1 Parallelogram (Piece F)
All 7 pieces combine without overlapping to form figures such as arrows, birds, running persons, and letters!
Domino Tiling of m × n Grids: The Area Parity Rule
Can an m × n grid of unit squares be tiled using 2 × 1 domino tiles?
Both dimensions are even. Each column of 4 squares can be covered by 2 vertical dominoes.
One dimension (4) is even. Each column of 4 squares can be covered by 2 vertical dominoes.
Total area is 35 (odd). Each 2×1 domino covers exactly 2 squares. An odd number of squares can never be covered by dominoes without a gap!
The Checkerboard Colouring Proof (Textbook Pages 158–160)
Click any square in the 5 × 3 grid to remove it, and inspect the Black vs White count
A 5 × 3 grid has 15 squares. When one square is removed, 14 squares remain (an even number!). Is it automatically tileable? Click a square below to remove it and find out:
Both colors have 7 squares! Equal counts remove the color obstruction, enabling a valid domino tiling.
If you try for hours and fail to find a tiling, that only means you haven't found one yet. But the black-and-white coloring argument is a mathematical proof of impossibility: it proves that no possible arrangement can ever exist!
Tiling the Entire Plane: The 360° Vertex Sum Condition
Which regular polygons can tile a flat surface seamlessly without gaps or overlaps?
3 × 120° = 360°. Exactly 3 regular hexagons meet at every vertex. Used by bees in honeycombs to maximize area with minimal perimeter!
In 2023, mathematicians discovered a 13-sided polygon nicknamed ‘the hat’—an aperiodic monotile that tiles the entire plane without ever repeating periodically!
Bees and wasps construct hexagonal cells because regular hexagons have the lowest perimeter-to-area ratio among plane-tiling polygons, maximizing honey storage with minimum wax!
Practice Zone (40 Graded Questions)
Progress through Foundation, Application, Challenge, and Master tiers
What is the mathematical definition of 'bisection' of a line segment?
When constructing the perpendicular bisector of segment XY with compass centres at X and Y, what must be true about the radius chosen?
What is the measure of the angle formed between a line segment and its perpendicular bisector?
How can a 45° angle be constructed using only a ruler and compass?
Which triangle congruence condition justifies that the angle bisector ray OC divides ∠XOY into two equal angles?
What type of triangle is formed when constructing a 60° angle at vertex A with equal arc radii AB = BC = AC?
What is the definition of 'tiling' (or tessellation) of a region?
Can a 5 × 7 rectangular grid of unit squares be tiled using 2 × 1 domino tiles?
How many pieces make up a traditional Chinese Tangram puzzle?
In the ancient Indian Śulba-Sūtras, what practical tool was used as both a compass and a straightedge?
When constructing the perpendicular bisector of segment XY, is it necessary that the radius of the arcs drawn above XY equals the radius of the arcs drawn below XY?
To construct a 90° angle at a given point O on a line, why is only ONE pair of intersecting arcs needed instead of two?
How can an angle of 30° be constructed using a ruler and compass?
How can an angle of 15° be constructed?
In the textbook angle-sum puzzle (Page 152), the angles around point O are 40°, 60°, 50°, 30°, 40°, and 90°. Will a 70° angle fit into the remaining gap?
Why do six congruent equilateral triangles arranged around a common vertex form a regular hexagon?
To construct a line parallel to line m through an external point B, which geometric principle is used?
A 4 × 7 grid has 28 unit squares. Can it be tiled using 2 × 1 dominoes?
In a checkerboard (black-and-white) colouring of a grid, why does every 2 × 1 domino cover exactly 1 black and 1 white square?
A 5 × 3 grid has 15 squares. If one corner square is removed, 14 squares remain. When colored as a checkerboard, there are 8 white squares and 6 black squares. Can the remaining 14 squares be tiled with 2 × 1 dominoes?
What is the locus of all points in a plane that are equidistant from two given points X and Y?
Can an angle of 65.5° be constructed using only an unmarked ruler and compass?
In the 6-pointed star construction (Textbook Page 154), six triangles are built on the edges of a regular hexagon. Are these six outer triangles equilateral?
If a region has an EQUAL number of black and white squares on a checkerboard, does this GUARANTEE that it can be tiled with 2 × 1 dominoes?
Why can regular hexagons tile the entire plane seamlessly, while regular octagons (8-sided) cannot tile the plane by themselves?
How do you construct a perpendicular to a line l through an external point P not lying on l?
What is the order of rotational symmetry of the 6-pointed star shown on textbook page 154?
What general rule determines whether an m × n rectangular grid of unit squares can be tiled with 2 × 1 dominoes?
In the trefoil arch construction (page 149), what geometric symmetry condition must the support lines AB and CD satisfy with the base segment AD?
Why do honeybees build honeycomb cells in regular hexagonal prisms rather than square or triangular prisms?
In the 8-petalled flower construction, what is the exact angle between any two consecutive supporting rays emanating from the center?
A line segment AB has length 10 cm. How many lines in the plane can be perpendicular bisectors of AB?
In 2023, mathematicians discovered a single non-periodic shape that can tile the entire plane without ever repeating periodically. What nickname was given to this 'einstein' tile?
If point P lies on the perpendicular bisector of segment XY, which of the following is strictly TRUE?
Consider an 8 × 8 chessboard with two diagonally opposite corner squares removed (leaving 62 squares). Can this board be tiled with 31 dominoes of size 2 × 1?
What is the interior angle of a regular polygon with n sides?
Which Dutch graphic artist (1898–1972) is world-famous for creating mathematical tilings featuring interlocking birds, fish, and reptiles?
When bisecting an angle ∠XOY, if the arcs of equal radius from A and B are drawn on the opposite side of O (away from the angle opening), does the resulting line OC still bisect ∠XOY?
Which of the following sets contains ALL regular polygons that can tile the Euclidean plane on their own?
In a regular hexagon with side length s, what is the distance between two opposite vertices (e.g. AD in Fig. 6.12)?
Final Chapter Assessment (20 Questions)
Comprehensive test of constructions, triangle congruence, angles, dominoes, and tessellations
Quick Revision Flashcards
What is bisection?
12 Common Geometric & Tiling Misconceptions to Avoid
Key Geometric & Tiling Terms
The division of a line segment, angle, or geometric quantity into two congruent (identical) halves.
A straight line that bisects a segment at its midpoint at an angle of exactly 90°.
AB ⟂ XY and OX = OYThe set of all points satisfying a given geometric condition. For a segment, the locus of points equidistant from both endpoints is its perpendicular bisector.
PX = PY ⟺ P lies on the perpendicular bisectorA ray that divides an angle into two angles of equal measure.
∠BOC = ∠AOC = ½ ∠AOBA triangle with all three sides of equal length and all three interior angles equal to 60°.
a = b = c, ∠A = ∠B = ∠C = 60°A six-sided polygon with all six sides equal and all six interior angles equal to 120°.
Interior angle = 120°, 6 equilateral trianglesThe property of a figure that looks identical to itself after a rotation by an angle strictly less than 360°.
Order n = 360° ÷ θAncient Indian Vedic geometric texts ('rules of the cord') describing altar constructions using ropes and pegs.
The covering of a flat plane or region using a collection of shapes with no gaps and no overlapping regions.
A rectangular tile composed of two congruent unit squares joined edge-to-edge (size 2 × 1).
The evenness or oddness of an integer. In domino tilings, since each tile covers 2 squares, the total area must be even.
Area = 2k (must be even)An impossibility proof technique where grid squares alternate black and white. Since each 2×1 domino covers 1 black and 1 white square, unequal counts ($B \ne W$) prove tiling is impossible.
B = W is necessary for domino tilingAn ancient Chinese geometric dissection puzzle formed by cutting a square into 7 geometric shapes.
An architectural arch featuring three symmetrical circular lobes supported by equal line segments.
A single geometric tile discovered in 2023 that can tile the entire flat plane without ever repeating in a periodic pattern.
Frequently Asked Questions (FAQ)
What is a perpendicular bisector and how is it constructed?
A perpendicular bisector is a straight line that divides a line segment into two equal halves at a 90° right angle. It is constructed using a compass by drawing two arcs of equal radius (greater than half the segment) from each endpoint above and below the segment, then joining their intersection points.
Why does the perpendicular bisector pass through the midpoint?
Every point on the perpendicular bisector is equidistant from the two endpoints. By triangle congruence (SSS on the outer triangles, followed by SAS on the half-triangles), the two segments formed on the line are proven equal in length (OX = OY) and the angles are equal straight-line supplementary angles (90° each).
How do you construct a 90° angle at a point on a line?
Mark two points X and Y at equal distances from the given point O using a compass, making O the midpoint of segment XY. Then construct the perpendicular bisector of XY. Since O is already on the bisector, you only need one pair of intersecting arcs above O, and joining that intersection to O gives the 90° angle.
What were the Śulba-Sūtras and how did they construct perpendiculars?
The Śulba-Sūtras are ancient Indian geometric treatises of the Vedic period that provided precise instructions for building fire altars. Instead of compasses, geometers used a rope with loops at the ends and a marked midpoint. By fastening the loops to pegs at endpoints X and Y and stretching the midpoint taut above and below, they created the perpendicular bisector.
How do you bisect any given angle using a ruler and compass?
Draw an arc centered at the vertex O cutting the two arms at points A and B (so OA = OB). From centres A and B, draw intersecting arcs of equal radius inside the angle to meet at point C. Ray OC bisects the angle into two equal parts because ΔOBC ≅ ΔOAC by SSS.
How do you copy an angle to another location without measuring it?
Draw an arc from the original vertex cutting the arms at two points. With the same compass radius, draw an arc at the new vertex. Measure the straight-line distance (chord) between the two arm intersections on the original angle with your compass, and transfer this length onto the new arc. Drawing a ray through the intersection reproduces the angle by SSS congruence.
How can a 60° angle be constructed with ruler and compass?
Draw an arc from vertex A cutting the baseline at B. Keeping the exact same compass radius, draw an arc from B intersecting the first arc at point C. Ray AC forms an equilateral triangle ΔABC with the baseline, so ∠CAB = 60°.
How do you construct 30° and 15° angles?
First construct a 60° angle using an equilateral triangle. Bisect the 60° angle to obtain two 30° angles. Bisecting a 30° angle yields a 15° angle.
How do you construct a regular hexagon using an unmarked ruler and compass?
A regular hexagon consists of six congruent equilateral triangles around a common central point. Construct a circle with radius equal to the desired side length. Step the compass around the circumference six times using the same radius to mark the six vertices, then connect adjacent vertices.
What is the mathematical definition of tiling (tessellation)?
Tiling is the complete covering of a region or flat plane using a set of shapes without any gaps (empty spaces) and without any overlapping boundaries.
Why can't a 5 × 7 grid be tiled using 2 × 1 dominoes?
A 5 × 7 grid contains 35 unit squares. Each 2 × 1 domino covers exactly 2 squares. Covering the board with dominoes would require 35 ÷ 2 = 17.5 dominoes, which is impossible because you cannot use half a domino. The area is odd, so tiling is impossible.
How does checkerboard (black-and-white) colouring prove that some regions cannot be tiled?
When a grid is colored like a checkerboard, every 2 × 1 domino must cover exactly one black square and one white square. If a region has unequal numbers of black and white squares (e.g. 8 white and 6 black), no domino tiling can ever exist.
Which regular polygons can tile the entire flat plane on their own?
Only three regular polygons can tile the Euclidean plane on their own: equilateral triangles (interior angle 60°, 6 meet at each vertex), squares (interior angle 90°, 4 meet at each vertex), and regular hexagons (interior angle 120°, 3 meet at each vertex). Their interior angles divide 360° evenly.
Why do bees construct honeycombs with hexagonal cells?
Regular hexagons tile the plane seamlessly without wasted space. Furthermore, among the three shapes that tile the plane (triangles, squares, hexagons), hexagons have the smallest perimeter for a given area, allowing bees to store the maximum honey while using the minimum amount of beeswax.
What is an aperiodic monotile, and when was one discovered?
An aperiodic monotile is a single geometric shape that can tile the entire plane without ever forming a repeating periodic pattern. Mathematicians searched for one for decades until 2023, when a shape nicknamed 'the hat' was discovered by an international team of mathematicians.
Chapter Summary: What You Built & Discovered!
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