Constructions and Tilings
Geometry is also about making and patterning shapes. This Class 7 Ganita Prakash chapter uses a ruler and compass to construct perpendiculars, parallel lines and triangles accurately, and explores tilings — how copies of a shape can cover a plane with no gaps or overlaps.
Learning objectives
- Construct perpendicular and parallel lines.
- Construct triangles from given measures.
- Understand tilings (tessellations).
- Identify shapes that tile the plane.
Key concepts
Tools of construction
Accurate geometric drawings use a ruler (for straight lines and lengths) and a compass (for arcs and equal lengths). A construction relies only on these tools, not on measuring angles with a protractor, so the steps must create the required figure exactly using arcs.
Perpendiculars and parallels
Using a compass, we can drop a perpendicular from a point to a line, or raise one at a point on a line, by making equal arcs. Parallel lines are constructed by copying an angle so that corresponding angles are equal, which guarantees the new line never meets the original.
Constructing triangles
A triangle can be constructed when enough measurements are given: three sides (SSS), two sides and the included angle (SAS), or two angles and the included side (ASA). The same information that proves congruence also determines a unique triangle to construct.
Tilings (tessellations)
A tiling, or tessellation, covers a flat surface with copies of one or more shapes leaving no gaps or overlaps. Squares, equilateral triangles and regular hexagons tile by themselves because their angles meet to make 360° around each point, which is why honeycombs are hexagonal.
Important formulas
Tiling condition
shapes meeting at a point have angles summing to 360°
Triangle construction
needs SSS, SAS or ASA information
Key definitions
- Compass
- A tool for drawing arcs and marking equal lengths.
- Perpendicular
- A line meeting another at a right angle (90°).
- Tessellation
- A tiling that covers a plane with no gaps or overlaps.
- Regular polygon
- A polygon with all sides and angles equal.
Solved examples
Q1. Which tool draws arcs in a construction?
Solution: The compass.
Q2. Name three regular shapes that tile the plane by themselves.
Solution: Equilateral triangle, square and regular hexagon.
Q3. What information lets you construct a unique triangle?
Solution: SSS, SAS or ASA.
Common mistakes to avoid
- Using a protractor where a compass construction is required.
- Changing the compass width between arcs that should be equal.
- Thinking every shape can tile the plane.
- Trying to construct a triangle from too little information.
Constructions and Tilings — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
Arcs in a construction are drawn with a:
Practice questions
Short answer
Which two tools are used for constructions?
A ruler and a compass.
What is a tessellation?
A tiling that covers a plane with no gaps or overlaps.
Name a regular polygon that does not tile the plane.
A regular pentagon.
Long answer
Explain why squares, equilateral triangles and regular hexagons tile the plane.
A shape tiles the plane when copies of it can surround each meeting point completely, with their angles adding up to a full turn of 360° and leaving no gaps or overlaps. A square has 90° angles, and four of them meet to make 360°. An equilateral triangle has 60° angles, and six meet to make 360°. A regular hexagon has 120° angles, and three meet to make 360°. Because each of these angles divides evenly into 360°, the shapes fit perfectly around every point — which is why floors are often tiled with squares and honeycombs are built from hexagons. A regular pentagon, with 108° angles, does not divide 360° evenly, so it cannot tile by itself.
What information is needed to construct a unique triangle, and how does this relate to congruence?
A triangle can be constructed exactly when we are given one of the sets of measurements that also guarantee congruence: three sides (SSS), two sides and the angle between them (SAS), or two angles and the side between them (ASA). Each of these fixes the triangle's shape and size completely, so there is only one possible triangle to draw. This is the same idea as congruence: any two triangles built from the same SSS, SAS or ASA information must be identical. If too little is given — for example only the three angles — the shape is fixed but the size is not, so a unique triangle cannot be constructed.
HOTS (Higher Order Thinking)
Can a regular octagon tile the plane on its own? Explain.
No. Its interior angle is 135°, which does not divide 360° evenly, so octagons alone leave gaps (they tile only when combined with squares).
Why can't a triangle be constructed from its three angles alone?
Because equal angles fix only the shape, not the size — infinitely many triangles of different sizes share the same angles.
Quick revision
Revision notes
- Constructions use only a ruler and compass.
- Perpendicular = meets at 90°; parallels never meet.
- Construct triangles from SSS, SAS or ASA.
- Shapes tile when angles around a point total 360°.
Key takeaways
- A compass makes accurate arcs and equal lengths.
- SSS, SAS, ASA each fix a unique triangle.
- Squares, triangles and hexagons tile the plane.
Frequently asked questions
What tools are allowed in a construction?
Only a ruler and a compass.
What is a tessellation?
A pattern of shapes covering a plane with no gaps or overlaps.
Why do hexagons tile the plane?
Three 120° angles meet to make 360° around each point.