Symmetry · Tessellations · Congruence · Constructions · 3D Shapes
Geometry surrounds us everywhere — from the honeycomb pattern of a beehive to the symmetrical wings of a butterfly, from the tiled floors in our homes to the three-dimensional shapes of buildings and boxes. In this chapter, we embark on a fascinating exploration of several geometric themes that connect the flat world of 2D shapes with the solid world of 3D objects.
The NCERT Ganita 2025-26 textbook brings together multiple geometric ideas under one roof, helping you see how symmetry, tessellations, congruence, constructions, and 3D geometry are all interconnected. Understanding these themes builds a strong foundation for higher-level geometry in Classes 9 and 10.
| Topic | Key Concepts |
|---|---|
| Symmetry | Line symmetry, rotational symmetry, order of rotation |
| Tessellations | Tiling patterns, regular and semi-regular tessellations |
| Congruence of Triangles | SSS, SAS, ASA, RHS criteria |
| Geometric Constructions | Bisecting angles, perpendicular bisector |
| Properties of Triangles | Angle sum property, exterior angle theorem |
| 3D Shapes | Faces, edges, vertices, Euler's formula |
| Nets of 3D Shapes | Unfolding solids, visualizing 3D in 2D |
A figure has line symmetry if there is a line (called the line of symmetry or axis of symmetry) that divides the figure into two parts such that one part is the mirror image of the other. When you fold the figure along this line, the two halves match exactly.
A figure has rotational symmetry of order n if it maps onto itself exactly n times during a full 360° rotation. The angle of rotation is 360° ÷ n.
| Shape | Lines of Symmetry | Rotational Order | Angle of Rotation |
|---|---|---|---|
| Equilateral Triangle | 3 | 3 | 120° |
| Square | 4 | 4 | 90° |
| Regular Pentagon | 5 | 5 | 72° |
| Regular Hexagon | 6 | 6 | 60° |
| Circle | Infinite | Infinite | Any angle |
| Parallelogram (non-rectangle) | 0 | 2 | 180° |
A tessellation (or tiling) is a pattern of shapes that covers a flat surface completely without any gaps or overlaps. You see tessellations everywhere — in bathroom tiles, brick walls, honeycomb structures, and decorative art.
For a regular polygon to tessellate the plane by itself, its interior angle must be a factor of 360°. Let us check:
| Regular Polygon | Interior Angle | 360° ÷ Angle | Can Tessellate? |
|---|---|---|---|
| Equilateral Triangle | 60° | 6 (whole number) | Yes |
| Square | 90° | 4 (whole number) | Yes |
| Regular Pentagon | 108° | 3.33... (not whole) | No |
| Regular Hexagon | 120° | 3 (whole number) | Yes |
| Regular Heptagon | 128.57° | 2.8 (not whole) | No |
| Regular Octagon | 135° | 2.67 (not whole) | No |
Two triangles are said to be congruent if they have exactly the same shape and size. When two triangles are congruent, all their corresponding sides are equal and all their corresponding angles are equal. We write △ABC ≅ △DEF.
However, we do not need to check all six measurements (3 sides + 3 angles). There are shortcut rules called congruence criteria:
SAS: Two sides (5 cm, 6 cm) and the included angle (50°) are equal ⇒ △ABC ≅ △DEF
Geometric constructions are drawings made using only a compass and a straight edge (unmarked ruler). No measurements are read from the ruler — we only use it to draw straight lines. This ancient practice trains precision and logical thinking.
To bisect angle ∠ABC means to draw a ray BD that divides the angle into two equal parts, so ∠ABD = ∠DBC.
The perpendicular bisector of a line segment AB is a line that passes through the midpoint of AB and is perpendicular to it. Every point on this line is equidistant from A and B.
The sum of the three interior angles of any triangle is always 180°. This is one of the most fundamental results in Euclidean geometry.
An exterior angle of a triangle is formed by one side of the triangle and the extension of an adjacent side. The Exterior Angle Theorem states:
A 3D shape (solid) has three dimensions — length, width, and height. Every polyhedron (a solid with flat faces) has faces (flat surfaces), edges (line segments where two faces meet), and vertices (corner points where edges meet).
The great mathematician Leonhard Euler discovered a beautiful relationship connecting faces (F), vertices (V), and edges (E) of any convex polyhedron:
| 3D Shape | Faces (F) | Vertices (V) | Edges (E) | F + V − E |
|---|---|---|---|---|
| Tetrahedron | 4 | 4 | 6 | 4 + 4 − 6 = 2 |
| Cube | 6 | 8 | 12 | 6 + 8 − 12 = 2 |
| Triangular Prism | 5 | 6 | 9 | 5 + 6 − 9 = 2 |
| Square Pyramid | 5 | 5 | 8 | 5 + 5 − 8 = 2 |
| Pentagonal Prism | 7 | 10 | 15 | 7 + 10 − 15 = 2 |
| Octahedron | 8 | 6 | 12 | 8 + 6 − 12 = 2 |
| Dodecahedron | 12 | 20 | 30 | 12 + 20 − 30 = 2 |
| Icosahedron | 20 | 12 | 30 | 20 + 12 − 30 = 2 |
A net is a 2D pattern that can be folded to form a 3D shape. Imagine cutting open a cardboard box and laying it flat — that flat layout is the net of the box. Nets help us understand how 3D shapes are constructed from flat surfaces.
When we draw a 3D shape on paper (a 2D surface), we use techniques to give the illusion of depth:
How many lines of symmetry does a regular hexagon have? What is its order of rotational symmetry?
Can regular octagons alone tessellate the plane?
In △ABC and △DEF, AB = DE = 5 cm, BC = EF = 7 cm, and ∠B = ∠E = 60°. Are the triangles congruent?
In △PQR, ∠P = 45° and ∠Q = 70°. Find ∠R and the exterior angle at R.
A polyhedron has 8 faces and 6 vertices. How many edges does it have?
In right triangles △ABC and △PQR, ∠B = ∠Q = 90°, hypotenuse AC = PR = 10 cm, and AB = PQ = 6 cm. Prove congruence.
Which of the following arrangements of 6 squares is a valid net of a cube: (i) a straight row of 6 squares, or (ii) a cross-shaped arrangement (1-4-1)?
Does a parallelogram (that is not a rectangle) have line symmetry? What about rotational symmetry?
What shapes can you get by slicing a cylinder with a flat plane?
SSS (Side-Side-Side): If all three sides of one triangle are equal to the three sides of another, the triangles are congruent. Example: △ABC with AB = 3 cm, BC = 4 cm, CA = 5 cm, and △DEF with DE = 3 cm, EF = 4 cm, FD = 5 cm. By SSS, △ABC ≅ △DEF.
SAS (Side-Angle-Side): If two sides and the included angle of one triangle equal those of another, the triangles are congruent. Example: AB = DE = 5 cm, ∠A = ∠D = 60°, AC = DF = 7 cm. By SAS, △ABC ≅ △DEF.
ASA (Angle-Side-Angle): If two angles and the included side of one triangle equal those of another, the triangles are congruent. Example: ∠B = ∠E = 50°, BC = EF = 6 cm, ∠C = ∠F = 70°. By ASA, △ABC ≅ △DEF.
RHS (Right-Hypotenuse-Side): If both triangles are right-angled, and the hypotenuse and one side of one triangle equal those of the other, the triangles are congruent. Example: ∠B = ∠Q = 90°, AC = PR = 13 cm, BC = QR = 5 cm. By RHS, △ABC ≅ △PQR.
Why SSA is not valid: When you know two sides and a non-included angle, two different triangles can sometimes be formed (the ambiguous case). The third vertex can be at two different positions, creating two non-congruent triangles with the same SSA measurements.
Given: △ABC with interior angles ∠A, ∠B, and ∠C.
To Prove: ∠A + ∠B + ∠C = 180°.
Construction: Through vertex A, draw a line PQ parallel to BC (PQ ∥ BC).
Proof:
Since PQ ∥ BC and AB is a transversal:
∠PAB = ∠B (alternate interior angles) ... (i)
Since PQ ∥ BC and AC is a transversal:
∠QAC = ∠C (alternate interior angles) ... (ii)
Now, ∠PAB + ∠BAC + ∠QAC = 180° (angles on a straight line PQ)
Substituting from (i) and (ii): ∠B + ∠A + ∠C = 180°.
Hence proved: ∠A + ∠B + ∠C = 180°.
Euler's formula: For any convex polyhedron, F + V − E = 2, where F = number of faces, V = number of vertices, and E = number of edges.
Example 1 (Tetrahedron): F = 4, V = 4, E = 6. Check: 4 + 4 − 6 = 2 ✔
Example 2 (Cube): F = 6, V = 8, E = 12. Check: 6 + 8 − 12 = 2 ✔
Example 3 (Square Pyramid): F = 5, V = 5, E = 8. Check: 5 + 5 − 8 = 2 ✔
Finding a missing value: If a polyhedron has 20 faces and 12 vertices, find E. Using F + V − E = 2: 20 + 12 − E = 2, so E = 30. This is an icosahedron.
Euler's formula is powerful because knowing any two of F, V, E lets you calculate the third. It holds for all convex polyhedra but may fail for shapes with holes (non-convex or non-simply-connected solids).
Steps to construct the perpendicular bisector of segment AB:
1. Open the compass to more than half the length of AB.
2. With A as centre, draw arcs above and below the line.
3. Without changing the compass width, with B as centre, draw arcs intersecting the first arcs at points P and Q.
4. Draw the straight line through P and Q. This is the perpendicular bisector of AB.
Why every point on it is equidistant from A and B:
Let M be the midpoint of AB (where the perpendicular bisector meets AB), and let X be any point on the perpendicular bisector.
In △XMA and △XMB:
• XM = XM (common side)
• MA = MB (M is the midpoint)
• ∠XMA = ∠XMB = 90° (perpendicular bisector)
By SAS criterion, △XMA ≅ △XMB.
By CPCT, XA = XB. Hence every point on the perpendicular bisector is equidistant from A and B.
A tessellation is a pattern of one or more shapes that covers a plane completely without gaps or overlaps.
Why only three regular polygons work alone:
At each vertex of a tessellation, the angles meeting at that point must sum to exactly 360°. For a regular polygon to tessellate by itself, its interior angle must divide 360° exactly (giving a whole number of polygons at each vertex).
• Equilateral triangle: 60° → 360/60 = 6 triangles at each vertex ✔
• Square: 90° → 360/90 = 4 squares at each vertex ✔
• Regular hexagon: 120° → 360/120 = 3 hexagons at each vertex ✔
• Regular pentagon: 108° → 360/108 = 3.33... ✘
• Regular heptagon: 128.57° → 360/128.57 = 2.8 ✘
For polygons with more than 6 sides, the interior angle exceeds 120°, and you cannot fit even 3 of them around a point (3 × 120 = 360, 3 × 121 > 360).
Semi-regular tessellations use two or more types of regular polygons, arranged so the same pattern of polygons surrounds every vertex. Examples: (i) Octagons + squares (135 + 135 + 90 = 360); (ii) Hexagons + triangles (120 + 60 + 60 + 60 + 60 = 360). There are exactly 8 semi-regular tessellations.