NCERT Solutions for Class 7th Maths Chapter 6 Regular Hexagons · Regular Hexagon and Equilateral Triangles — In-text Questions

Book page 151 Updated on2026-09-19

Q1.

How do we construct a regular pentagon (5-sided figure) and a regular hexagon (6-sided figure)? To begin with, try to construct a pentagon and hexagon with equal sidelengths.

Answer

Making all the sides equal is easy with a compass. Making all the angles equal as well is the hard part — and that is what "regular" demands.

Regular polygon	Sides	Each angle	Ruler and compass?
Equilateral triangle	3	60°	Yes — done in Grade 6
Square	4	90°	Yes — done in Grade 6
Regular pentagon	5	108°	Needs more theory — later years
Regular hexagon	6	120°	Yes — from 60°, in this chapter

With a compass you can quickly draw a five-sided or six-sided closed figure with equal sides: mark off one length, turn the ruler a little, mark it again, and keep going until the shape closes. But when you measure the angles you will find they are not equal, so the figure is not regular.

Why it happens: equal sides do not force equal angles. A rhombus has four equal sides but is not a square. For a regular polygon we need to control the angle at each corner, and to do that with only a ruler and compass we must be able to construct that angle. 120° comes from 60°, which comes from an equilateral triangle — so the hexagon is within reach. 108° cannot be reached from 60° and 90° by bisecting, so the pentagon must wait.

Q2.

Can we break a regular hexagon into smaller pieces that can be constructed?

Answer

Yes — into six equilateral triangles.

Join the centre O to each of the six vertices A, B, C, D, E, F.
The hexagon splits into 6 triangles: ΔOAB, ΔOBC, ΔOCD, ΔODE, ΔOEF, ΔOFA.
Angle of each triangle at O = 360° ÷ 6 = 60°
OA = OB = … = OF (all radii of the same circle), so each triangle is isosceles
An isosceles triangle with a 60° apex has base angles (180° − 60°) ÷ 2 = 60°
⇒ each triangle is equilateral

Why it happens: equilateral triangles are the one polygon we already know how to construct exactly, straight from the compass. If a hexagon is nothing but six of them stuck together, then constructing the hexagon reduces to constructing an equilateral triangle six times — a problem we have already solved.

Q3.

What happens when we join the ‘opposite’ points of a regular hexagon? Since a regular hexagon has equal sides and angles, can we expect a figure like this? Will all the triangles in the figure be equilateral triangles?

Fig. 6.12, page 151.

Answer

The three long diagonals AD, BE and CF all pass through one point O, and they cut the hexagon into six equilateral triangles.

The three main diagonals of a regular hexagon meet at O and cut it into six equilateral triangles, each with a 60° angle at O.

Six equal angles fill the turn at O:
each = 360° ÷ 6 = 60°
All six segments OA, OB, …, OF are equal ⇒ each triangle is isosceles
Isosceles + 60° apex ⇒ other two angles = (180° − 60°) ÷ 2 = 60° each
⇒ all six triangles are equilateral

Also, the hexagon's angle at each vertex = 60° + 60° = 120° ✓

Why it happens: a regular hexagon is as symmetric as a shape can be — turn it by 60° and nothing changes. That turning symmetry forces the six segments from the centre to be equal and the six angles at the centre to be equal. Equal radii plus a 60° angle is exactly the recipe for an equilateral triangle.

Check it yourself: the book asks us to argue the other way round, which is safer — start with six equilateral triangles, show they fit around a point, and see that a regular hexagon comes out. That is done in the next questions.

Q4.

Can six congruent equilateral triangles be placed together as in Fig. 6.12? If yes, will it result in a regular hexagon?

Fig. 6.12, page 151.

Answer

Yes to both. Six 60° angles exactly fill the turn around a point, and the outer boundary is a regular hexagon.

Angle of each equilateral triangle = 60°
Six of them at the centre: 60° × 6 = 360°
360° is the complete angle around a point
⇒ no gap is left and no two triangles overlap ✓

Sides of the resulting figure: each is a side of one triangle,
and the triangles are congruent ⇒ all six sides are equal ✓
Angle at each outer vertex = 60° + 60° = 120°, the same at all six ✓
Equal sides + equal angles ⇒ regular hexagon

Why it happens: a degree was defined by taking the full turn around a point to be 360°. So a set of angles can be packed around a point without gap or overlap exactly when they add up to 360°. Six 60° angles do; that is the whole reason a hexagon works and, for example, a regular pentagon does not (5 × 108° = 540°, and 108° does not divide 360° evenly either).

Tip: this "angles around a point add to 360°" test is the same test used later in the chapter to decide which regular polygons can tile the plane.

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