Constructions and Tilings

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CLASS VII Mathematics ~6 marks/year Ch 14 of 15
Constructions and Tilings

Class 7 · Mathematics · NCERT chapter notes · Akanksha Classes

Snapshot
  • This chapter has two halves: geometric constructions using only a ruler (straightedge) and compass, and tilings (tessellations) — covering the plane with shapes that leave no gaps and no overlaps.
  • You learn to construct a circle, copy and bisect a line segment, draw the perpendicular bisector, and construct special angles like $60°$, $30°$, $90°$ and $45°$.
  • You construct triangles from given measurements and explore which shapes can and cannot be made.
  • A tiling works because the angles meeting at every corner (vertex) add up to exactly $360°$.
  • Regular polygons that tile by themselves are only the equilateral triangle, square and regular hexagon.
  • You study line symmetry and rotational symmetry in tiling patterns, and meet beautiful real-life tilings (floors, honeycombs, Islamic art).
  • Weightage: ~6 marks/year — a construction with steps (3–4 marks) plus 1–2 short tiling/symmetry questions.
Detailed Notes

1. The tools of construction

In geometry, a construction means drawing a figure accurately using only two tools: a ruler used as a straightedge (to draw straight lines, not to measure unless told) and a compass (to draw circles and mark equal lengths). A sharp pencil and a protractor (for checking) complete your kit.

A construction must be exact, not a rough sketch. Always leave the faint compass arcs visible — they are the proof of your method.

The power of the compass is that it carries equal distances. Whatever its opening, every arc it draws is the same radius from the centre. This single fact underlies all the constructions below.

2. Constructing a circle

A circle is the set of all points at a fixed distance (the radius) from a fixed point (the centre).

Construction — circle of radius 4 cm

1. Mark a point O for the centre. 2. Open the compass to $4$ cm using a ruler. 3. Place the metal point at O and turn the pencil all the way round. The closed curve is the required circle. Any line from O to the curve is a radius ($4$ cm); a line through O joining two points on the curve is a diameter ($8$ cm).

3. Copying a line segment

Construction — copy segment AB onto a ray

1. Draw any ray with starting point P. 2. Open the compass to the length AB. 3. With the point at P, cut an arc on the ray at Q. Then PQ $=$ AB. The compass simply transferred the exact length.

4. Perpendicular bisector of a segment

The perpendicular bisector of AB is the line that cuts AB into two equal halves at $90°$. Every point on it is equidistant from A and B.

Construction — perpendicular bisector of AB

1. Draw segment AB. 2. Open the compass to more than half of AB. 3. With centre A, draw arcs above and below AB. 4. With the same radius and centre B, draw two more arcs cutting the first pair at points P (top) and Q (bottom). 5. Join PQ. The line PQ is the perpendicular bisector; it meets AB at its midpoint M and makes a right angle there.

Why it works: P and Q are each the same distance from A and from B, so the line through them must be the set of equidistant points — exactly the perpendicular bisector.

5. Constructing a $60°$ angle (and $30°$)

Construction — angle of 60°

1. Draw a ray OA. 2. With centre O, draw an arc cutting OA at P. 3. With the same radius and centre P, draw an arc cutting the first arc at Q. 4. Join OQ. Then $\angle$AOQ $=60°$.

Why: OP, OQ and PQ are all equal (same compass radius), so triangle OPQ is equilateral and each of its angles is $60°$.

To get $30°$, simply bisect the $60°$ angle.

6. Bisecting an angle

Construction — bisect ∠AOB

1. With centre O, draw an arc cutting OA at X and OB at Y. 2. With centre X, draw an arc in the interior; with the same radius and centre Y, draw another arc cutting it at Z. 3. Join OZ. Then OZ divides $\angle$AOB into two equal angles.

7. Constructing $90°$ and $45°$

A right angle can be built from $60°$ and a bisected step, or as the perpendicular at a point on a line.

Construction — 90° using 60° and 120°

Make a $60°$ arc and a $120°$ arc on the same baseline, then bisect the angle between them. The bisector gives $\dfrac{60°+120°}{2}=90°$. Bisecting a $90°$ angle then gives $45°$.

Quick angle ladder: $60°$ (equilateral arc) → bisect for $30°$; build $90°$ → bisect for $45°$; bisect $45°$ for $22.5°$.

8. Constructing triangles

A triangle is fixed once enough sides/angles are known. Common cases:

  • SSS — all three sides given.
  • SAS — two sides and the angle between them.
  • ASA — two angles and the side between them.
Construction — SSS triangle, sides 5 cm, 4 cm, 3 cm

1. Draw base BC $=5$ cm. 2. With centre B and radius $4$ cm, draw an arc. 3. With centre C and radius $3$ cm, draw an arc cutting the first at A. 4. Join AB and AC. Triangle ABC is complete. (Here the arcs do meet because $4+3>5$.)

Triangle inequality: a triangle is possible only if the sum of any two sides is greater than the third. With sides $2,3,7$ the arcs never meet ($2+3<7$) — no triangle exists.

9. What is a tiling (tessellation)?

A tiling or tessellation covers a flat surface using one or more shapes so that there are no gaps and no overlaps, repeating forever. Bathroom floors, brick walls and honeycombs are everyday tilings.

The key rule: at every meeting point (vertex) of a tiling, the angles around that point must add up to exactly $360°$ (a full turn).

10. Which regular polygons tile by themselves?

A regular polygon has all sides and all angles equal. The interior angle of a regular $n$-gon is $\dfrac{(n-2)\times180°}{n}$. For copies of one regular polygon to tile, its interior angle must divide $360°$ exactly.

Regular polygon Interior angle $360°\div$ angle Tiles?
Triangle (3)$60°$$6$Yes
Square (4)$90°$$4$Yes
Pentagon (5)$108°$$3.33\dots$No
Hexagon (6)$120°$$3$Yes
Only three regular polygons tile the plane alone: the equilateral triangle, the square, and the regular hexagon. The pentagon fails because $108°$ does not divide $360°$.
Worked example — why the hexagon tiles

Three regular hexagons meet at a vertex: $120°+120°+120°=360°$ — a perfect full turn, no gap, no overlap. That is exactly why honeycombs are hexagonal.

11. Tilings with more than one shape

Even shapes that cannot tile alone may tile when combined, as long as the angles at each vertex still total $360°$. For example, regular octagons ($135°$ each) and squares ($90°$) fit together: $135°+135°+90°=360°$. Such patterns are common on old floors. Also, every triangle and every quadrilateral (even irregular ones) can tile the plane — a surprising and useful fact.

Worked example — any quadrilateral tiles

The four angles of any quadrilateral add to $360°$. By rotating copies, all four different angles can be brought to meet at one point, summing to $360°$ and covering the plane.

12. Symmetry in tilings

Tilings are rich in symmetry:

  • Line (reflection) symmetry: the pattern looks the same on both sides of a mirror line.
  • Rotational symmetry: the pattern looks the same after turning it through a certain angle about a point. A square tiling has rotational symmetry of order $4$ (it matches every $90°$ turn).
  • Translation symmetry: sliding the whole pattern by the right distance lands it exactly on itself — this is what makes a tiling "repeat".

Artists and architects use these symmetries to create stunning patterns, from Indian jaali screens to the tile work of the Alhambra.

13. Common mistakes to avoid

  • Changing the compass radius midway through a bisection — keep it fixed for both arcs.
  • Erasing the construction arcs — they show the method and earn marks.
  • Trying to tile with a regular pentagon alone — $108°$ does not divide $360°$.
  • Forgetting the triangle inequality before constructing — check the arcs will meet.
  • Confusing line symmetry (mirror) with rotational symmetry (turn).

14. Quick revision checklist

  • Constructions use only ruler (straightedge) and compass; keep arcs visible.
  • Perpendicular bisector: equal arcs from both endpoints, then join the crossings.
  • $60°$ comes from an equilateral triangle arc; bisect to halve any angle.
  • Triangle exists only if sum of any two sides $>$ third side.
  • Tiling rule: angles at each vertex total $360°$.
  • Only triangle, square and hexagon tile alone among regular polygons.
  • Symmetry types: reflection, rotation, translation.
Practice MCQs
1. The two basic tools of geometric construction are:
  1. Protractor and set square
  2. Ruler (straightedge) and compass
  3. Compass and protractor
  4. Divider and scale
Answer: (B) A straightedge for lines and a compass for arcs/equal lengths.
2. Every point on the perpendicular bisector of AB is:
  1. nearer to A
  2. nearer to B
  3. equidistant from A and B
  4. on segment AB only
Answer: (C) That equidistance is exactly what defines the perpendicular bisector.
3. The angle built directly from a single equilateral-triangle arc is:
  1. $30°$
  2. $45°$
  3. $60°$
  4. $90°$
Answer: (C) Equal radii make an equilateral triangle, whose angles are $60°$.
4. To construct $30°$ you should:
  1. bisect a $60°$ angle
  2. bisect a $90°$ angle
  3. bisect a $45°$ angle
  4. draw a $30°$ arc directly
Answer: (A) Half of $60°$ is $30°$.
5. A triangle with sides $2$ cm, $3$ cm and $7$ cm:
  1. can be constructed
  2. cannot be constructed
  3. is right-angled
  4. is equilateral
Answer: (B) $2+3=5<7$, so the arcs never meet (triangle inequality fails).
6. At every vertex of a tiling, the angles must add up to:
  1. $90°$
  2. $180°$
  3. $270°$
  4. $360°$
Answer: (D) A full turn, so there are no gaps or overlaps.
7. Which regular polygon does NOT tile the plane by itself?
  1. Equilateral triangle
  2. Square
  3. Regular pentagon
  4. Regular hexagon
Answer: (C) Its interior angle $108°$ does not divide $360°$ exactly.
8. How many regular hexagons meet at a vertex in a hexagonal tiling?
  1. $2$
  2. $3$
  3. $4$
  4. $6$
Answer: (B) $120°\times3=360°$.
9. The interior angle of a square is $90°$, so the number of squares meeting at a vertex is:
  1. $2$
  2. $3$
  3. $4$
  4. $6$
Answer: (C) $360°\div90°=4$.
10. Six equilateral triangles meet at a vertex because each angle is:
  1. $45°$
  2. $60°$
  3. $72°$
  4. $90°$
Answer: (B) $60°\times6=360°$.
11. When bisecting an angle with a compass, between the two interior arcs you must:
  1. change the radius
  2. keep the same radius
  3. use a protractor
  4. draw a circle
Answer: (B) Equal radii guarantee the two halves are equal.
12. A pattern that looks the same after a $90°$ turn about a point has:
  1. line symmetry only
  2. rotational symmetry
  3. no symmetry
  4. translation only
Answer: (B) Matching under a turn is rotational symmetry.
13. Regular octagons and squares tile together because $135°+135°+90°=$
  1. $270°$
  2. $315°$
  3. $360°$
  4. $405°$
Answer: (C) The vertex angles sum to a full $360°$.
14. Which statement is TRUE?
  1. No quadrilateral can tile the plane
  2. Every quadrilateral can tile the plane
  3. Only squares can tile the plane
  4. Only rectangles can tile the plane
Answer: (B) The four angles of any quadrilateral add to $360°$, so copies can tile.
15. The line that cuts a segment into two equal parts at a right angle is called its:
  1. diameter
  2. median
  3. perpendicular bisector
  4. tangent
Answer: (C) "Perpendicular" (right angle) + "bisector" (equal halves).
Important Questions
Q1. Describe, step by step, how to construct the perpendicular bisector of a line segment AB. (3 marks)
Answer: Draw AB. Open the compass to more than half of AB. With centre A draw arcs above and below; with the same radius and centre B draw arcs cutting them at P and Q. Join PQ — it bisects AB at its midpoint and meets it at $90°$. P and Q are equidistant from A and B, which is why PQ is the perpendicular bisector.
Q2. Construct an angle of $60°$ and explain why the method works. (3 marks)
Answer: Draw ray OA. With centre O draw an arc cutting OA at P. With the same radius and centre P, cut the arc at Q. Join OQ; $\angle$AOQ $=60°$. Since OP $=$ OQ $=$ PQ (same radius), triangle OPQ is equilateral and each angle is $60°$.
Q3. State the rule for a tiling and explain why a regular pentagon cannot tile the plane on its own. (2 marks)
Answer: The angles meeting at each vertex must total $360°$. A regular pentagon's interior angle is $108°$, and $360°\div108°=3.33\dots$ is not a whole number, so copies cannot fit around a point without a gap or overlap.
Q4. Name the three regular polygons that tile the plane by themselves and give the number meeting at each vertex. (3 marks)
Answer: Equilateral triangle ($6$ meet, $6\times60°=360°$), square ($4$ meet, $4\times90°=360°$), and regular hexagon ($3$ meet, $3\times120°=360°$).
Q5. Explain how to construct a triangle with sides $5$ cm, $4$ cm and $3$ cm, and state the condition for a triangle to be possible. (3 marks)
Answer: Draw base BC $=5$ cm. With centre B radius $4$ cm and centre C radius $3$ cm, draw arcs that meet at A; join AB and AC. The triangle is possible because the sum of any two sides exceeds the third ($4+3>5$, $5+3>4$, $5+4>3$) — the triangle inequality.
Q6. Name and describe the three types of symmetry found in tiling patterns. (3 marks)
Answer: Line (reflection) symmetry — the pattern matches across a mirror line; rotational symmetry — it matches after turning through some angle about a point; translation symmetry — sliding the pattern by a fixed distance lands it on itself, which makes it repeat.
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Large Numbers Around Us · Arithmetic Expressions · A Peek Beyond the Point · Expressions using Letter-Numbers · Parallel and Intersecting Lines · Number Play · A Tale of Three Intersecting Lines · Working with Fractions
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