NCERT Solutions for Class 8 Maths Chapter 11 Exploring Some Geometric Themes (PDF) – 2027

Start with one square. At every step, cut each square that is still present into 9 equal sub-squares and rub out the middle one.

Yes. The number of squares that remain is multiplied by 8 at every step, and the number of holes goes up by exactly the number of squares that were present before the step.

Yes — R_n = 8ⁿ.

Add up the new holes made at every earlier step: H_n = 1 + 8 + 8² + … + 8ⁿ⁻¹.

Let ABC be equilateral with side a, and let P, Q, R be the midpoints of BC, CA and AB.

So the middle triangle PQR has sides a⁄2, a⁄2, a⁄2 as well. All four triangles are equilateral with side a⁄2, so they are identical.

Begin with a filled equilateral triangle. At every step, join the midpoints of each remaining triangle and remove the middle one.

Triangles remaining: 1, 3, 9, 27, … — that is 3ⁿ. Holes: 0, 1, 4, 13, … — that is 1 + 3 + 3² + … + 3ⁿ⁻¹.

Each remaining triangle gives 3 remaining triangles and 1 new hole at the next step.

Carpet: (8⁄9)ⁿ sq. units. Gasket: (3⁄4)ⁿ sq. units.

Sierpinski Carpet. Each square is cut into 9 equal parts, so a square of Step n has area (1⁄9)ⁿ, and there are 8ⁿ of them.

Sierpinski Gasket. Each triangle is cut into 4 identical parts, so a triangle of Step n has area (1⁄4)ⁿ, and there are 3ⁿ of them.

Start with an equilateral triangle. Replace every straight side by the four-segment ‘bump’: divide the side into three equal parts, build an equilateral triangle on the middle part, and rub out that middle part.

S_n = 3 × 4ⁿ.

P_n = 3 × (4⁄3)ⁿ units.

Hold the written name in your mind as a picture, then read the picture from right to left.

A square is left — standing on its corner, with half the area of the original.

To make another square, put the four corner triangles together with their right angles meeting at one point. Their long sides (hypotenuses) then form the four sides of a new square, the same size as the square left in the middle.

A regular hexagon.

An octagon — eight sides, but not a regular one.

Take the square with side 3, so the marks are 1 unit apart. Each corner cut removes a right isosceles triangle with legs 1, so its hypotenuse is √2.

A cube, seen straight on so that one face is parallel to the wall.

A sphere — and from every viewpoint, which no other solid manages.

• Cylinder — looked at along its axis (end on).
• Cone — looked at from directly below (or above) along its axis.
• Hemisphere — looked at along its axis of symmetry.

A cone, seen from the side.

• Cone — from the side the outline is an isosceles triangle: the base circle appears as a segment and the two slant edges as the other two sides.
• Square pyramid or triangular pyramid — from the side.
• Triangular prism — looked at along its length, the outline is the triangular face itself.

A cylinder — a tin of ghee, a candle, a piece of chalk.

A cone — a party hat, a heap of grain, a road-work cone.

A hemisphere will not do — from the side it gives a semicircle, not a triangle. The straight slant edges of the cone are what make the side profile triangular.

A triangular prism — the shape of a glass prism, or of a tent.

A square pyramid also works: from directly above it is a square (a special rectangle), and from the side it is a triangle.

A frustum of a cone — a cone with its top sliced off parallel to the base. A bucket, a matka, a lampshade or a tumbler has this shape.

A pentagonal prism — the shape of a five-sided pillar or a pentagon-shaped box.

Another answer: a house-shaped solid — a cuboid with a triangular prism roof on top. From the front it gives a pentagon (a square with a triangular cap); from the side, a rectangle.

Not unique — every one of these conditions is met by many different solids.

A decagonal prism has 12 faces, 30 edges and 20 vertices.

For an n-sided polygon, count the same way:

A decagonal pyramid has 11 faces, 20 edges and 11 vertices.

For an n-sided base:

A net of a cube is the flat shape you get by cutting the cube along some of its edges and unfolding it onto a plane — six squares joined edge to edge, which can be folded back up to give the cube.

Fold along every internal line, all in the same direction, keeping one square flat on the table.

1. Keep the third square of the long strip (the one with a neighbour on each side) flat — this becomes the base.
2. Fold the left and right squares up. They become two opposite walls.
3. Fold the square below the base up. It becomes a third wall.
4. Fold the strip going upwards: the square just above the base becomes the fourth wall, and the one above that folds over to become the lid.

(ii), (iii), (iv) and (vi) are nets of a cube; (i) and (v) are not.

• (i) No — both extra squares are on the same side of the row of four, so one of them folds onto a face already covered and one face is left open.
• (ii) Yes — two rows of three.
• (iii) Yes — the staircase.
• (iv) Yes — one square above and one below the row of four.
• (v) No — both extra squares hang below the same column; they overlap.
• (vi) Yes — one square above and one below the row of four.

The 11 nets of a cube:

Use the same ‘1 – 4 – 1’ layout as the cube net, but now the six rectangles come in three matching pairs.

(i) 5 cm × 3 cm × 1 cm. Lay a horizontal strip of four rectangles that wraps round the 5 cm length, then attach the two 3 cm × 1 cm ends.

(ii) 6 cm × 3 cm × 2 cm. The same layout: two rectangles 6 × 3, two 6 × 2 and two 3 × 2.

A net of a regular tetrahedron is a flat shape of four equilateral triangles, joined edge to edge, that folds up into the tetrahedron.

The 1st (the big triangle) and the 3rd (the strip of four) are nets. The 2nd and the 4th are not.

• 2nd: the fourth triangle hangs from an edge that gets joined to another edge when the other three triangles fold up, so it overlaps a face and one face stays open.
• 4th: it has 5 triangles, but a tetrahedron has only 4 faces.

No — a regular tetrahedron has only 2 nets: the big triangle and the strip of four triangles. Every other way of joining four triangles is one of these turned or flipped over, or does not fold up.

Take the edge to be 6 cm. Then a net is a single equilateral triangle of side 12 cm, with its three midpoints joined.

How to draw it accurately. Draw a 12 cm segment. With the compass set at 12 cm, cut arcs from both ends; their meeting point is the third vertex. Join up, mark the midpoints with a ruler and join them.

Take a square base of side 6 cm and slant edges of 8 cm. The net is that square with an isosceles triangle built outwards on each of its four sides.

Draw each triangle with compasses: from both ends of a side of the square, cut arcs of 8 cm; they meet at the tip of the triangle.

Two circles and one rectangle. Unroll the curved surface after cutting it once along the height, and lay the two circular ends flat.

One side is the height h of the cylinder; the other is the circumference of the base, 2πr.

A circle for the base and a sector of a larger circle for the curved surface.

The sector has radius equal to the slant height l of the cone, and its arc is exactly as long as the base circle, 2πr.

A sector of a circle with centre O, where O is the apex of the cone.

Every point on the rim of the base is at the same distance l (the slant height) from O. Unrolling does not change these distances, so the curved edge of the net is an arc of a circle with centre O and radius l.

You get a slanting (oblique) cone — a cone that leans to one side, with its tip not above the centre of its base.

In such a net the boundary points are at different distances from O, so after rolling, the slanting lines from the tip to the rim have different lengths — long on one side, short on the other — and the cone tilts.

Take a prism whose ends are equilateral triangles of side 5 cm and whose length is 9 cm.

An octahedron has 8 equilateral triangles as faces, so every net has 8 triangles. The net shown is a strip of 6 triangles (pointing up and down in turn) with one triangle above and one below the strip.

To visualise the folding: the strip of 6 rolls into a ring round the middle of the octahedron, and the two extra triangles close its top and its bottom.

Use equilateral triangles of side 5 cm.

1. Draw a strip of 6 equilateral triangles of side 5 cm, pointing up and down in turn.
2. Add one triangle on the top edge of the 4th triangle and one on the bottom edge of the 3rd triangle, exactly as in the book’s net.
3. Cut it out, adding small flaps on some outer edges for gluing.
4. Fold along every inner line and tape the free edges in pairs: the strip closes into a ring and the two extra triangles close its ends.

It cannot be done. A sphere has no net.

A cylinder and a cone flatten out because they are curved in only one direction — every point of them lies on a straight line drawn on the surface. A sphere carries no straight lines at all; it curves in every direction at once. So paper, which will bend but not stretch, can never lie flat against it.

The shortest path is the straight line joining the ant and the laddu on the net. Unfold the end face (with the ant) so that it lies flat next to the top face (with the laddu) and join the two points with a straight line.

Folded back, this path goes straight up the end face to the middle of the top edge, then straight along the middle of the top face to the laddu.

Again, unfold and join the ant and the laddu by a straight line. With the end face laid flat next to the top face, the straight red line runs slantwise across the end face and the top face to the middle of the front edge. This is the shortest path (the red path in the book).

By turning the question into one we can already answer. Unfold the box; the path becomes a path on a flat net, with its length unchanged. On a plane we know that the straight line is shortest — so if our path unfolds into a straight line, nothing can beat it.

Only the first one is. Draw the net and see what each path looks like on it.

The test is simple: unfold, and look at whether the path comes out straight. A path that still has a kink in it on the flat net can always be shortened by pulling it taut.

It shows that the problem of the shortest path on a cuboid is the same problem as the shortest path on its net.

No, not yet — and the next two pages show exactly what is missing.

We have proved that the shortest path must be a straight line on some net. But a cuboid can be unfolded in many different ways, and:

• on some nets the straight segment between the two points runs outside the net, so it does not correspond to any path on the box at all;
• different nets give straight segments of different lengths.

In the first net the straight line leaves the net, so unfold another way: lay the end face (ant) flat next to the front face (laddu on its bottom edge).

Unfold the cuboid a different way — one that lays out the faces the ant would actually walk over, so that the straight segment stays inside the net.

1. Decide which faces the path could cross.
2. Unfold so that exactly those faces lie flat, side by side, in the order the ant would meet them.
3. Join the two points with a straight line and check that it stays inside the net.
4. Measure it with the Baudhayana Theorem.
5. Repeat for every other reasonable set of faces, and keep the smallest length.

The shortest path is 40 cm.

Unfold the box so that the ant’s path crosses the top face, a side face and the bottom face (the book’s second unfolding), and join the ant and the laddu with a straight line. It is the longest side of a right triangle:

The straight route along the bottom is 1 + 30 + 11 = 42 cm, which is longer.

It never gets longer. The projection of a line segment is shorter than the segment, or equal to it when the segment is parallel to the plane.

p ≤ l, always.

AECD is a rectangle, so AE = DC = p. Triangle AEB is right-angled at E, so AB = l is its hypotenuse, the longest side.

Exactly when the segment is parallel to the plane (or already lying in it).

In that case ABCD is itself a rectangle, so DC = AB exactly.

The projection of a square can be a square, a rectangle, a parallelogram, or a line segment — but never anything else.

The projection of a parallelogram is always another parallelogram (or, in the flattest case, a line segment). It can never be a quadrilateral that is not a parallelogram.

This is because parallel lines project to parallel lines, so both pairs of opposite sides stay parallel.

So a trapezium, a kite or an ordinary quadrilateral can never be the shadow of a parallelogram.

It is again a convex polygon with n sides — but it is normally not regular. It is regular only when the polygon is parallel to the plane.

Following the hint: the projection of the polygon is built from the projections of its n sides. Each side projects to a segment, corners project to corners, and the pieces stay joined in the same order. So the picture is an n-sided polygon (unless the polygon is perpendicular to the plane, when it flattens to a segment).

Cube (Fig. 4.4). Held with one face parallel to the plane, its projection is a square — the near face and the far face land exactly on top of each other, and the four side faces project onto the edges.

Cone (Fig. 4.5). Held with its axis parallel to the plane, its projection is a triangle — the base circle flattens to a segment (the base of the triangle) and the two extreme slant lines give the other two sides.

Yes — in every case there are many others. Fig. 4.6 shows two families of examples.

Held with its axis parallel to the plane, a cone projects to an isosceles triangle. So does each of these:

• A triangular prism whose end face is that triangle, seen along its length.
• A square pyramid of the same height and base width, seen from the side.
• A flat triangular sheet of card of exactly that shape, held parallel to the plane.
• A hollow cone, or a cone of any material at all — the projection cannot tell.

Yes. All three lines have the same front view. The lines lie flat (horizontal), so the top view shows each line at its full length, and

Fix each solid in its natural position — flat faces parallel to the planes, axes vertical — and read off the three outlines.

A shadow on the wall whose shape is very close to the projection of the object on that wall — but usually larger, and sometimes stretched or slightly distorted.

Bring the torch closer and the shadow grows; move it back and the shadow shrinks, settling down towards the size of the object’s outline and never getting smaller than that.

Because the light leaves the torch as a cone of spreading rays, not as a parallel beam.

Every ray that grazes the edge of the object carries on and lands further out on the wall. The closer the torch is to the object, the more steeply those edge rays are spreading, and the further out they land — so the shadow is bigger. Pull the torch back and the rays reaching the object are much more nearly parallel, so the outline they carve out on the wall is much closer to the object’s true outline.

Solids numbered (i)–(vi) left to right, top to bottom. In the side views the front of the solid is on the left; in the top views the front is at the bottom.

(i) 3 cubes: two in front, one behind the right-hand cube.

(ii) 4 cubes: an arm two cubes long with a column of two beside its back cube.

(iii) 4 cubes: a row of three at the back, one in front of the left end.

(iv) 14 cubes: a 3 × 3 base with a stepped wall on its back row.

(v) 4 cubes, one cube deep: a column of three with one cube on the right of the middle cube.

(vi) 13 cubes: a 3 × 3 base with its front-right cube missing and a U-shaped wall on its back row.

(i) The C is only 1 cube deep, so from the side it is a column of 4 squares and from the top a row of 3 squares.

(ii) Add a flat A-shaped layer (3 wide, 4 deep) behind the C’s top row: 7 more cubes, 15 in all. They hide behind the top row, so the front view stays C.

(iii) Yes. Add 2 cubes going back from the cube above the C’s bottom-left corner. The side view becomes F (17 cubes).

(iv) Yes, e.g. L, L, L (three bars from one corner), or any letter one cube thick (plain bars from the top and side).

Solid (ii).

Read the plan from the views. The top view shows the floor plan (L-shaped, front-right missing); the front and side views give the heights:

Solid (ii) has exactly this shape: a tall step at the back left, a lower step beside it, a single layer in front, and the front-right corner cut away (no slot or hole).

The nine views come in three rows — top, front and side of one solid each. One solid for each row:

(Numbers are the heights of the stacks, left to right; 0 means an empty place.)

20 cubes — 10 that can be seen and 10 hidden underneath.

Each row further back is one cube higher, and a cube cannot float, so every cube we see stands on a full column of cubes:

Three families of shapes: a square, a rectangle, and a hexagon — including a regular hexagon.

Balance the cube on one corner so that the long diagonal through that corner runs straight up and down. Now project it down onto the floor.

The reason is symmetry. Look at the three edges meeting at the bottom corner. Spin the cube through 120° about the vertical diagonal: the cube lands exactly on itself, and those three edges swap round among themselves. So no one of them is in any way special — they must all be tilted from the vertical by the same angle, and therefore their shadows on the floor must all be the same length.

Every edge of the cube is parallel to one of these three edges, so all twelve edges project to the same length.

The five flat shapes are the ways of joining four squares edge to edge (counting a shape and its turned-over copy as the same):

How to draw them. On the isometric grid, use the three directions | for height, one slant for depth and the other slant for length. Then draw one cube at a time, or better, draw the whole outline by counting edges as you go.

Draw it exactly like the 1 × 1 × 1 cube, but make every edge 2 grid units long instead of 1.

1. Draw the outline: a hexagon with every side 2 units long.
2. Draw the three inner edges, each 2 units long, from the centre of the hexagon to the bottom corner and to the two upper side corners.
3. To show the small cubes, join the midpoints of opposite edges on each visible face. Shade the top, left and right faces differently if it helps.

Because two things that could have gone wrong do not.

Parallel stays parallel. The chapter proved that projection takes parallel lines to parallel lines. A cuboid has three families of parallel edges — height, length, depth — so on the paper they become three families of parallel lines: |, and the two slants. Every edge on the paper therefore announces which axis of the solid it belongs to, just by its direction.

Equal stays equal. Because the projection is isometric, a unit step along any of the three axes becomes a step of the same length on the paper. So one grid unit means one cube, whichever direction you count in.

Yes — take them one at a time, and for each one first decide the orientation.

Yes — three more, making 8 in all. The five in Fig. 4.8 are the flat ones; the extra three cannot be laid flat on a table, because they reach into all three directions.

Sample drawings on the isometric grid:

Draw edge by edge: vertical grid lines for the height, one slanting direction for the length and the other for the depth. Count one grid unit per cube, and for each edge decide whether it goes up or down (forwards or backwards) along its axis. Draw lightly first and darken only the edges that can be seen.

Yes. Following the arrows, the ball climbs up two steps and never goes down, yet it comes back to where it started. That is impossible on a real staircase — the picture is an illusion like the famous ‘endless staircase’.

Any small part of the drawing (for example the front row and the two steps) is an ordinary solid, with the three directions — height, length, depth — easy to see. Only when the loop is closed does it become impossible.

To recreate it, draw the realistic part first along the three grid directions, then continue the top row back along the depth direction and join it to the start, as in the picture. On the paper the lines meet, although in space the two ends would be at different heights.

(i) No. Its three arms run along the three different axes (up, along the length, along the depth), each at right angles to the others, and such arms can never close into a triangle. A model built from cubes always has a gap at one corner; it only looks closed from one special viewpoint.

Profiles of the model that can be built (three arms with a gap at one corner):

(ii) Draw one arm of cubes along each of the three grid directions, as in the picture, and draw the last arm so that on the paper its end meets the start of the first.

(iii) An isometric drawing shows no depth. Two ends that are far apart in space can land on the same point of the paper, and our eyes assume they are joined.