NCERT Solutions for Class 5 Maths Chapter 7 Shapes and Patterns (PDF) – 2027

Follow the eight steps and you get a chequered mat — it looks like a chessboard, with the two colours swapping places in every row.

Yes. Weave Row 1 as 2 over, 1 under again and again, and Row 2 as 2 under, 1 over again and again. Then Row 3 is like Row 1, Row 4 is like Row 2, and so on.

Here is the pattern written out band by band, for a mat with 12 bands. Weave your strips exactly like this.

(O = the strip passes over that band  ·  U = the strip passes under it.)

Row 5 is the same as Row 1, Row 6 as Row 2, and so on — the design repeats after every 4 rows.

Yes. Sample answer — the purple mat (16 bands):

The completed design (21 squares by 21 squares):

1. Border: shade every other square all round the edge.
2. Three diamond outlines, one inside the other, with a gap of one square between them. The given part is the top of the outer two.
3. Centre: a small solid diamond of 13 squares (1, 3, 5, 3, 1 in five rows).

No, a fourth pentagon will not fit. The empty space left is far too thin for it.

Leaving the gap would leave a hole in the floor; pushing a fourth pentagon in makes it overlap the others. So regular pentagons do not tessellate.

Yes, they fit perfectly. Six triangles fit around one point.

The six corners fill the whole turn round the point — no gap and no overlap. So equilateral triangles tessellate.

Yes, squares fit together with no gap and no overlap. You need 4 squares.

Every corner of a square is a right angle, and four right angles make one full turn round the point.

No. Five squares cannot fit around a point. The fifth one has to climb on top of the others.

Four squares already fill the whole turn round the point, so there is no room left for a fifth.

Yes. Three regular hexagons fit around a point.

Three hexagon corners fill the whole turn round the point, with no gap and no overlap — like a honeycomb.

Regular hexagons and equilateral triangles.

Look at the picture in the book. The big orange shapes have six equal sides — those are regular hexagons. The blue shapes between them have three equal sides — those are equilateral triangles.

Keep adding a purple hexagon with six yellow triangles around it, again and again. The little piece drawn in the book is the start of a hexagon-and-triangle tiling.

No. Regular octagons do not tessellate. Wherever two of them meet, a square hole is left behind.

Two octagons side by side leave a square-shaped gap, and a third octagon is too big for that gap — it would overlap.

Sample answer: At each marked point, 2 squares and 3 triangles come together (once the pattern is continued). Yes, the same set of shapes meets at every marked point. To continue, keep repeating this group, then colour all squares one colour and all triangles another.

No, the triangles are not equilateral. Two sides of each triangle are equal — they are sides of the squares. The third side (red) is shorter: on the dot grid it goes straight across 2 spaces, while each slanting side goes 2 spaces across and 1 space up or down. So each triangle has only 2 equal sides.

Three triangles and two squares come together at each marked point (once the pattern is continued). Yes — the same five shapes meet at every marked point.

Walk round the big red dot in the picture above and count what you pass:

Sample answer — four patterns that work: (1) squares only — 4 meet at each point; (2) equilateral triangles only — 6 at each point; (3) regular hexagons and triangles — 2 hexagons and 2 triangles at each point; (4) squares and triangles — 2 squares and 3 triangles at each point. Regular pentagons alone will not work.

With 2 pieces you can make these shapes: two different isosceles triangles, a kite, a rectangle and two different parallelograms.

You can make 2 different types of triangle. Both of them have exactly 2 equal sides — they are isosceles triangles.

The two bottom angles land exactly on top of each other. So each isosceles triangle has 2 equal angles.

No — not with the pieces of this rhombus. Every triangle you build has exactly 2 equal sides, never 3.

With the rhombus in the book, the base of the taller triangle (the short diagonal) comes out slightly longer than its two equal sides. It is very close, so measure carefully.

Not by joining two pieces. Every triangle made from two pieces has 2 equal sides. (But each single piece is itself a triangle with all three sides unequal — a short side, a long side and a slanting side.)

Yes. All three angles of an equilateral triangle are equal.

Fold so that one corner lands on another: each time the two corners match exactly.

None. All three sides of each new triangle are different lengths. A triangle like that is called a scalene triangle.

No two angles of a scalene triangle are equal. All three angles are different.

Fold one corner onto another: for every pair, one corner sticks out past the other.

You can make 4 different four-sided shapes: a kite, a rectangle, and two different parallelograms. The book shows three of them.

A kite has two pairs of equal sides, and the equal sides are next to each other. Sides that are next to each other are called adjacent sides.

In both A and B, the sides that are equal are the opposite ones, not the adjacent ones.

In A the opposite angles are equal. In B all four angles are equal, and each one is a right angle — so B is a rectangle.

Use the corners of the grid squares as your points. Here are two kites and two parallelograms you can copy.

(b) A 4-sided shape and (c) a 5-sided shape can be made, as shown. (a) With the rhombus in the book, three pieces never close up into an exact triangle — a small step or gap is always left, so the outline has 4 or more sides.

(b) Rectangle, (c) Triangle, (d) Pentagon, (e) Hexagon and (f) Octagon can be made. Only (a) Square cannot.

A tangram is one square cut into 7 pieces. Here they are, with their names.

Same: five of the seven pieces are triangles, and all five are exactly the same shape — a right-angled triangle with two equal sides. Different: they come in three sizes, and two pieces are not triangles at all.

Every corner in the tangram set is a right angle, half a right angle, or one and a half right angles. Each triangle has one right angle and two equal corners, each half a right angle. The square has four right angles. The parallelogram has two corners of half a right angle and two of one and a half right angles.

Every triangle has 2 equal sides. The square has all 4 sides equal. The parallelogram has its opposite sides equal.

The four outlines in the book are a parallelogram, a rectangle, a triangle and a trapezium. Here is one way to make each.

1 Triangle — the two large triangles. 2 Parallelogram — the two large triangles, one turned half a turn. 3 Rectangle — the square and the two small triangles. 4 Trapezium — the square with a small triangle on each side.

Rectangle.

Rhombus.

Two shapes fit: the rhombus and the parallelogram.

Two shapes fit: the rhombus and the parallelogram.

Square.

All four shapes fit: the square, the rectangle, the rhombus and the parallelogram.

Two shapes fit: the rectangle and the square.

Yes. Statements 3, 4, 6 and 7 each describe more than one shape.

Follow the four steps and you get a kite — two equal short sides at the square corner, two equal long sides meeting at the tip, and the crease as a mirror line down the middle.

You can see the kite itself, and triangles inside it.

A rectangle is formed. Its opposite sides are equal and all four of its angles are right angles.

You get a rectangle every single time. Only the size and slant of the rectangle change, never the kind of shape.

No. Joining the ends of two diameters always gives a rectangle.

Keep repeating two moves: jump forward 10 points, then go back 9 points. After 48 straight lines you arrive back at point 1 and the design is complete.

Yes. Draw a middle circle, then four circles of the same size centred on its top, bottom, left and right points. Then, inside each outer circle, draw one more arc of the same size with the compass point on that circle’s far end — these four arcs cut off the outer caps.

Opposite faces:

(a) 7 small cubes  (b) 32 small cubes  (c) 81 small cubes.

(a) 8  (b) 12  (c) 6  (d) 1

The order, from left to right, is:

The names simply say how many faces the solid has.

Icosahedron: equilateral triangles.  Dodecahedron: regular pentagons.

Icosahedron: yes.  Dodecahedron: yes. In both solids every face is exactly the same shape and the same size.

Icosahedron: 5 faces meet at each vertex. Dodecahedron: 3 faces meet at each vertex. And yes — in both solids the number is the same at every single vertex.

Both solids have 30 edges.

The counting trick — count face by face, then halve

Yes — the cube, the tetrahedron and the octahedron. In all of them the same number of faces meets at every vertex.

Sample answer — the three models in the book are an octahedron, a triangular prism and a pentagonal prism.