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Ganita Manjari · Class 9 · Part II · Chapter 12

Ganita Manjari Class 9 Maths Chapter 12 Quadrilaterals Exercise 12.4 Solutions

Two questions on tiling the plane: why a regular pentagon cannot tile it, and how a non-convex 4-gon DART tiles it, with a drawn tiling.

Last updated 6 October 2026

  • 2Questions
  • 1Diagrams
  • Ch 12Quadrilaterals

How to answer every question

  1. Find an angle sum Use the sum of angles of a polygon: (n − 2) × 180°.
  2. Fill the full turn The angles at a vertex of a tiling add up to 360°.
  3. Try a half-turn Rotate a tile by 180° about the midpoint of a side so that it fits that side.
  4. Check at each vertex See that every angle of the tile appears once at a vertex.

Tiling: covering the plane with copies of a shape, with no gaps and no overlaps.

Regular polygon: all sides equal and all angles equal.

Half-turn: a rotation by 180° about a point.

Tilings

Chapter 12 Exercise 12.4 Question 1 Solution Justify why the plane cannot be tiled with a regular pentagon. (Hint: read the first 3 sentences of "Think and Reflect" in the section on tiling.)

Answer: 108° does not divide 360°
Step 1 · The angle of a regular pentagon

The angles of a pentagon add up to (5 − 2) × 180° = 540°. In a regular pentagon all five are equal:

each angle = 540°5 = 108°
Step 2 · Angles around a vertex

In a tiling, the tiles meeting at a vertex fill the full turn: their angles add up to 360°. If only regular pentagons meet there, k of them give 108° × k.

Number of pentagons kTotal angle 108° × kEqual to 360°?
2216°no
3324°no
4432°no, too large

We would need 108k = 360, so k = 103, which is not a whole number.

What about a vertex on the side of another tile?

There the angles must add up to 180° instead, and 108k = 180 gives k = 53, again not a whole number.

The angles at a vertex must add up to 360° (or 180° on a side), but 108° times a whole number is never 360° or 180°. So a regular pentagon cannot tile the plane.

Chapter 12 Exercise 12.4 Question 2 Solution Draw a non-convex 4-gon DART. Show how we can tile the plane with copies of DART. Both methods discussed earlier will work. Which do you prefer?

Answer: both methods work; Method 1 is easier
Method 1 · Half-turns about side midpoints
  1. Draw DART, say D(1.5, 1.5), A(0, 0), R(5, 0), T(1.5, 4); D is the dent.
  2. Take a copy and rotate it by 180° about the midpoint of a side. It fits exactly along that side.
  3. Repeat this for every side of every copy.
  4. Around each vertex four tiles meet, and they use the four angles of DART once each, so they fill 360°.
Tiling the plane with copies of the non-convex 4-gon DARTCopies of DART tile the plane without gaps. The yellow DART is the starting tile; blue tiles are translations of it and pink tiles are its 180-degree rotations about side midpoints.Q2 · tiling with DART: each new copy is a 180° turn about the midpoint of a side
Q2 · tiling with DART by half-turns about side midpoints
Method 2 · The Varignon grid

Draw the Varignon parallelogram of DART. Make a grid of copies of that parallelogram, place copies of DART in the matching positions (as in Exercise 12.3 Q5), and place the extra copies in the gaps. DART tiles the plane in the same way.

The reason this works for a non-convex 4-gon is that Theorem 9 and the Midpoint Theorem hold for non-convex quadrilaterals too.

Which do I prefer?

Method 1. Each step is the same simple move (turn about a midpoint), and no extra construction is needed. Method 2 first needs the Varignon parallelogram and its grid, and then needs careful placing of each copy.

Copies of DART tile the plane by half-turns about side midpoints (Method 1) or by using the Varignon grid (Method 2). I prefer Method 1 because it is a single repeated move.

Answers at a glance

Tilings are decided by the angles at a vertex.

QuestionWhat is askedKey ideaAnswer
Q1 Regular pentagonAngles at a vertex add to 360°108° does not divide 360°
Q2 Tile with DARTHalf-turns about side midpointsBoth methods work; Method 1 is easier

Other chapters of Ganita Manjari (Class 9)

Source of the questions: NCERT, Ganita Manjari, Grade 9, Part II, Chapter 12, Exercise Set 12.4. The solutions, explanations and diagrams on this page are our own working.

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