Chapter 4 exercise answers: Exploring Some Geometric Themes

Open NCERT p. 72One way to think about it

1. Step 0 is a solid (filled) equilateral triangle.
2. To get Step 1: join the midpoints of the three sides. This cuts the triangle into 4 identical smaller equilateral triangles. Remove the middle (upside-down) one, keeping the 3 corner triangles filled.
3. To get Step 2: apply the exact same rule to each of the 3 filled triangles left in Step 1 - join each one's midpoints and remove its middle triangle. You now have 9 small filled triangles arranged in the Step-1 pattern, with holes at 2 sizes.
4. Repeating this forever (Step 3, Step 4, ...), always feeding the rule its own output, is the Sierpinski Triangle.

In shortStep 0: one filled triangle. Step 1: 3 filled triangles (the middle one removed). Step 2: 9 filled triangles (the middle one removed from each of the 3). The count of filled triangles is 1, 3, 9, ... (multiplied by 3 every step).

Watch this explained “Two rules, side by side”, 0:00 into The Sierpinski carpet and gasket: what repeated removal leaves behind · हिंदी में देखें

Open NCERT p. 72Checked by computer

1. At Step 0 there is 1 triangle and 0 holes.
2. Every surviving triangle is cut into 4 and loses only its middle piece, so it leaves behind exactly 3 triangles. So the number of triangles is multiplied by 3 at every step: 1, 3, 9, 27, ... which is 3 raised to the power n.
3. Every surviving triangle also punches exactly 1 new hole at the next step, and no old hole ever closes up. So the new holes added at step n equal the number of triangles that were surviving at step (n-1), namely 3 raised to the power (n-1).
4. Adding up the holes made so far: 1 + 3 + 9 + ... + 3^(n-1). This is a geometric series that sums to (3^n - 1)/2.

AnswerTriangles remaining after step n: 3^n. Holes after step n: (3^n - 1)/2. For example step 3 has 3^3 = 27 triangles and (27-1)/2 = 13 holes.

Watch this explained “The same two sentences, with three for eight”, 6:00 into The Sierpinski carpet and gasket: what repeated removal leaves behind · हिंदी में देखें

Open NCERT p. 72Checked by computer

1. Start each fractal with area 1 sq. unit at Step 0.
2. Sierpinski Carpet: each step cuts every surviving square into 9 equal pieces and removes 1, so 8/9 of whatever area was left survives. After n steps the area is (8/9) raised to the power n.
3. Sierpinski Triangle: each step cuts every surviving triangle into 4 equal pieces and removes 1, so 3/4 of whatever area was left survives. After n steps the area is (3/4) raised to the power n.

AnswerCarpet: area after step n = (8/9)^n sq. units. Triangle: area after step n = (3/4)^n sq. units. Both shrink towards 0 as n grows, because 8/9 and 3/4 are both less than 1.

Watch this explained “The area was already inside the count”, 6:46 into The Sierpinski carpet and gasket: what repeated removal leaves behind · हिंदी में देखें

Open NCERT p. 73One way to think about it

1. Step 0 is a solid equilateral triangle (3 straight sides).
2. To get Step 1: on every side, mark the two points that divide it into 3 equal parts, raise an equilateral triangle outward on the middle third, and rub out that middle third. Each side turns into a 4-segment 'bump'. The triangle becomes a 6-pointed star with 3 x 4 = 12 sides.
3. To get Step 2: apply the very same 'divide into 3, raise a bump on the middle third' rule to each of the 12 sides of the Step-1 star. That gives 12 x 4 = 48 sides.
4. Continuing this forever gives the Koch Snowflake.

In shortStep 0: a triangle, 3 sides. Step 1: a 6-pointed star, 12 sides. Step 2: a more crinkled star, 48 sides. The side count is multiplied by 4 at every step.

Watch this explained “One side becomes a bump”, 0:00 into The Koch snowflake, and how its sides and perimeter grow at each step · हिंदी में देखें

Open NCERT p. 73Checked by computer

1. Step 0 is a triangle, so it has 3 sides.
2. Every single side, wherever it came from, is replaced by exactly 4 shorter sides at the next step (3 pieces from the cut, minus the removed middle, plus the 2 new sides of the raised triangle: 3 - 1 + 2 = 4).
3. Since every side turns into 4 sides independently, the total number of sides is multiplied by 4 at each step.
4. Starting from 3 sides and multiplying by 4, n times in a row, gives 3 x 4^n sides after step n.

AnswerNumber of sides after step n = 3 x 4^n. For example step 3 has 3 x 4^3 = 192 sides.

Watch this explained “Four times as many, every time”, 2:17 into The Koch snowflake, and how its sides and perimeter grow at each step · हिंदी में देखें

Open NCERT p. 73Checked by computer

1. The starting triangle has 3 sides of length 1 unit each, so its perimeter is 3 units.
2. Every side is cut into 3 equal parts, so at each step every side's length becomes a third of what it was.
3. The number of sides multiplies by 4 at each step (found in the previous question), while each side's length is multiplied by 1/3. So the perimeter (number of sides times length of one side) is multiplied by 4/3 at each step.
4. Starting from perimeter 3 and multiplying by 4/3, n times, gives 3 x (4/3)^n.

AnswerPerimeter after step n = 3 x (4/3)^n units. Since 4/3 is more than 1, the perimeter keeps growing without any limit as n increases, even though the whole snowflake stays inside a fixed circle.

Watch this explained “Count times length”, 3:37 into The Koch snowflake, and how its sides and perimeter grow at each step · हिंदी में देखें

Open NCERT p. 80Checked by computer

1. A cube has 6 faces, so a net must have 6 squares, and when it is folded each square must land on a different face. All six shapes have 6 squares, so the real test is: do two squares ever land on the same face?
2. (i) The row of 4 squares folds round into the 4 side walls of the cube. The square above the first square of that row folds over to close one end. The square to the left of it then folds down onto a side wall - the same wall as the last square of the row. Two squares on one face, and the other end is left open, so (i) is not a net.
3. (ii) Two rows of 3 squares that overlap in one column. Folding it (or rolling a cube across it one square at a time) puts the 6 squares on 6 different faces, so (ii) is a net.
4. (iii) The staircase of three pairs of squares also puts its 6 squares on 6 different faces, so (iii) is a net.
5. (iv) The row of 4 folds into the 4 side walls; the square above the third square closes one end and the square below the third square closes the other end. So (iv) is a net.
6. (v) The row of 4 folds into the 4 side walls. The square below the second square closes one end. The square below that then folds onto a side wall - the one already covered by the fourth square of the row. Two squares on one face and one end left open, so (v) is not a net.
7. (vi) The row of 4 folds into the 4 side walls; the square above the third square closes one end and the square below the second square closes the other. So (vi) is a net.

AnswerNets of a cube: (ii), (iii), (iv) and (vi). Not nets: (i) and (v), because in each of them two squares fold onto the same face.

Watch this explained “Six pieces - which of them fold?”, 3:13 into Nets: which flat shapes fold into which solid · हिंदी में देखें

Open NCERT p. 81One way to think about it

1. Every net has 6 squares. Sort the nets by their longest straight row of squares; turning a net round or flipping it over does not make a new net.
2. Longest row of 4: the row folds into the 4 side walls, and the other two squares must be one above the row and one below it (two on the same side would overlap). Number the squares of the row 1 to 4 from the left. The different nets are: above 1 and below 1; above 1 and below 2; above 1 and below 3; above 1 and below 4; above 2 and below 2; above 2 and below 3. That is 6 nets (every other choice is one of these turned or flipped).
3. Longest row of 3, with a pair on one side and a single square on the other (3 nets). Writing each row from top to bottom and numbering the columns from the left: (a) top: column 3; middle: columns 1 to 3; bottom: columns 3 and 4. (b) top: column 3; middle: columns 2 to 4; bottom: columns 1 and 2. (c) top: column 4; middle: columns 2 to 4; bottom: columns 1 and 2.
4. Longest row of 3, two rows of 3 (1 net): top: columns 3 to 5; bottom: columns 1 to 3.
5. Longest row of 2 (1 net), the staircase: top: columns 3 and 4; middle: columns 2 and 3; bottom: columns 1 and 2.
6. A row of 5 or 6 squares can never fold into a cube, because the row would wrap round more than the 4 side walls and overlap itself.
7. Total: 6 + 3 + 1 + 1 = 11. (Of all 35 ways of joining 6 squares edge to edge, exactly these 11 fold into a cube.)

In shortThe 11 nets are: 6 with a row of 4 (one square above the row and one below, at positions 1-1, 1-2, 1-3, 1-4, 2-2, 2-3), 3 with a middle row of 3 plus a pair on one side and a single square on the other, 1 made of two rows of 3 overlapping in one column, and 1 staircase of three pairs.

Watch this explained “Thirty-five tried, eleven fold”, 5:10 into Nets: which flat shapes fold into which solid · हिंदी में देखें

Open NCERT p. 81One way to think about it

(i) 5 cm, 3 cm, and 1 cm

1. A cuboid has 3 pairs of identical rectangular faces, one pair for every 2 of the 3 sidelengths: 5 cm x 3 cm, 3 cm x 1 cm, and 5 cm x 1 cm.
2. Arrange the 6 rectangles in a cross/strip so that every edge that is glued to another rectangle in the net has matching length: for example a middle row of 5x1, 5x3, 5x1, 5x3 rectangles side by side (sharing their 5 cm edges), with one 3x1 rectangle attached above and one 3x1 rectangle below the first 5x3 rectangle (sharing their 3 cm edges).

In shortAny layout of two 5x3, two 3x1 and two 5x1 rectangles that shares matching edge lengths and folds up without gaps is a correct net, for example the '1-4-1' cross described above.

(ii) 6 cm, 3 cm, and 2 cm

1. This cuboid's 3 pairs of faces are 6 cm x 3 cm, 3 cm x 2 cm, and 6 cm x 2 cm.
2. Use the same '1-4-1' cross layout: a middle row of 6x2, 6x3, 6x2, 6x3 rectangles side by side (sharing their 6 cm edges), with one 3x2 rectangle above and one 3x2 rectangle below the first 6x3 rectangle (sharing their 3 cm edges).

In shortAny layout of two 6x3, two 3x2 and two 6x2 rectangles that shares matching edge lengths and folds up without gaps is a correct net, for example the '1-4-1' cross described above.

Watch this explained “Six squares, six faces, one each”, 2:24 into Nets: which flat shapes fold into which solid · हिंदी में देखें

Open NCERT p. 92One way to think about it

1. In Fig. 4.6 the three lines start from the same point and all lie flat (none of them goes up or down). They differ only in how far they go back, away from the vertical plane.
2. Front view: it keeps the left-right length and the height and throws away the backward direction. All three lines go the same distance left to right and not at all up or down, so all three front views are the same line, of the same length.
3. Side view: it keeps the backward direction and the height. The first line does not go back at all, so its side view is just a point. The second goes back a little, so its side view is a short line; the third goes back further, so its side view is longer.
4. Top view: it keeps the left-right direction and the backward direction. Since the lines are flat, nothing is lost, so the top view of each line is exactly as long as the line itself - the longest of its three views.
5. In the top view, the line is the slanting side of a right triangle whose other two sides are the front-view length (left-right) and the side-view length (backward). So by the Baudhayana (Pythagoras) theorem: (top view length)^2 = (front view length)^2 + (side view length)^2.
6. For example, a flat line that goes 4 units across and 3 units back has front view 4, side view 3 and top view 5, and 5^2 = 4^2 + 3^2.

In shortYes. No view is ever longer than the line itself, and a view shows the full length only when the line is parallel to that plane. For these lines the front views are all equal, the top view shows the true length, and (top view length)^2 = (front view length)^2 + (side view length)^2.

Watch this explained “Never longer. Ever.”, 2:20 into Front, top and side views, and what each one loses · हिंदी में देखें

Open NCERT p. 92One way to think about it

1. The views depend on how the solid is placed, so first fix a position; a different position gives different views, and that is also a correct answer. Below, each solid stands on its natural base, and the views are listed as front, top, side.
2. Cube (resting on a face): square, square, square - all the same size.
3. Cuboid (resting on a face, edges along the three directions): three rectangles, one for each pair of edge lengths, in general all different.
4. Parallelepiped (a cuboid slanted to the left or right, resting on its base): the front view is a parallelogram, and the top and side views are rectangles.
5. Cylinder (standing on a circular end): rectangle (height by diameter), circle, rectangle.
6. Cone (standing on its circular base, tip up): triangle, circle with a dot at the centre for the tip, triangle.
7. Triangular prism (standing on a triangular end): rectangle, triangle, rectangle.
8. Square pyramid (standing on its square base, tip up): triangle, square with both diagonals drawn (the sloping edges meeting at the tip), triangle.

In shortOne set of answers (front, top, side): cube - square, square, square; cuboid - three rectangles; parallelepiped - parallelogram, rectangle, rectangle; cylinder - rectangle, circle, rectangle; cone - triangle, circle, triangle; triangular prism - rectangle, triangle, rectangle; square pyramid - triangle, square with diagonals, triangle. Placing a solid differently changes its views.

Watch this explained “Three planes, three views”, 5:51 into Front, top and side views, and what each one loses · हिंदी में देखें

Open NCERT p. 93Checked by computer

1. Number the pictures in each of the FRONT, TOP and SIDE columns 1 to 8 from the top. The rows of pictures are not in the same order as the objects on the left, so each object has to be found by what its outline shows.
2. Row 1 is the car (grille and headlamps from the front, roof from the top, wheels from the side).
3. Row 2 is the lidded pot, the last object on the left (a lidded pot from the front, a circle with the lid's handle from the top, the handle over the lid from the side).
4. Row 3 is the slide (a narrow strip with the box on it from the front, a long strip from the top, the ladder and ramp from the side).
5. Row 4 is the chair, row 5 is the fan (blades seen edge-on from the front and side, three blades from the top), row 6 is the funnel (a cone with a spout, and a circle with a dot from the top), and row 7 is the hammer (the head end-on from the front, the claw from the top, the full hammer from the side).
6. Row 8 is the cup with a pouring lip, the first object on the left (the V-shaped lip from the front, a circle with the lip and handle from the top, the handle from the side).
7. In this picture each object's front, top and side views all sit in the same row.

AnswerCup (1st object, green cup with a lip): row 8 in all three columns. Funnel: row 6. Hammer: row 7. Car: row 1. Slide: row 3. Chair: row 4. Fan: row 5. Lidded pot (8th object): row 2. Rows are counted 1 to 8 from the top of each column.

Watch this explained “Three planes, three views”, 5:51 into Front, top and side views, and what each one loses · हिंदी में देखें

Open NCERT p. 95One way to think about it

Top row, left

1. Convention used below: the front view is drawn as seen along the Front arrow, the top view as seen from above with the front edge at the bottom, and the side view as seen along the Side arrow (from the right), with the front edge on the left. 'Heights 1, 2' means columns of 1 square and 2 squares, from left to right.
2. This solid is 3 cubes, all on the ground: 2 cubes side by side at the front, and 1 more cube behind the right-hand one.

In shortFront view: 2 squares side by side. Top view: an L of 3 squares - 2 in the front row, 1 behind the right one. Side view: 2 squares side by side (front and back), 1 high.

Top row, right

1. This solid is 4 cubes: on the left, 2 cubes one behind the other on the ground; on the right, a column 2 cubes tall standing beside the back one.

In shortFront view: a single square on the left and a column of 2 squares on the right. Top view: an L of 3 squares - the left column 2 deep, and 1 square beside the back one. Side view: a single square at the front (left) and a column of 2 squares at the back (right).

Middle row, left

1. This solid is 4 cubes, all on the ground, in an L: a row of 3 cubes across the back and 1 cube in front of its left end.

In shortFront view: a row of 3 squares. Top view: an L of 4 squares - a row of 3 at the back and 1 square in front of the left end. Side view: 2 squares side by side (front and back), 1 high.

Middle row, right

1. This solid is a base 3 cubes wide, 3 cubes deep and 1 cube high (9 cubes), with a wall on its back row: the left two columns of the wall rise 2 more cubes, the right column 1 more cube (5 cubes). 14 cubes in all.

In shortFront view: 3 columns of heights 3, 3, 2. Top view: a 3 by 3 square. Side view: 2 columns of height 1 at the front and a column of height 3 at the back (an L).

Bottom row, left

1. This solid is 4 cubes standing up in one flat layer, 1 cube deep: a column of 3 cubes with 1 cube attached to the right of the middle cube.

In shortFront view: a column of 3 squares with 1 square beside the middle one. Top view: 2 squares side by side. Side view: a column of 3 squares.

Bottom row, right

1. This solid (13 cubes) has, from the front: a front row 2 cubes wide on the left, 1 high; a middle row 3 cubes wide, 1 high; and a back row 3 cubes wide whose columns rise to heights 3, 2 and 3 (a notch in the middle).

In shortFront view: 3 columns of heights 3, 2, 3 (a U-shaped top). Top view: 3 rows - back row 3 squares, middle row 3 squares, front row 2 squares on the left. Side view: 2 columns of height 1 at the front and a column of height 3 at the back (an L).

Watch this explained “Three planes, three views”, 5:51 into Front, top and side views, and what each one loses · हिंदी में देखें

Open NCERT p. 95One way to think about it

(i) What does it look like from the side? From the top?

1. The C is 3 cubes wide, 4 cubes tall and only 1 cube thick: a top row of 3, two cubes below its left end, and a bottom row of 3.
2. From the side you look along its width, so every row collapses to one square: you see a column of 4 squares.
3. From the top you look down its height and see only the top row: a row of 3 squares.

In shortFrom the side: a column of 4 squares (1 wide, 4 tall). From the top: a row of 3 squares (3 wide, 1 deep).

(ii) Glue additional cubes to make a shape that looks like …

1. The front view and the top view share the left-right width. The C is 3 squares wide, so the letter seen from the top must also be 3 squares wide. (The book's small picture of that letter is 4 squares wide; drawn exactly that size it could not share a width with the C, so it is built 3 wide here.)
2. Keep the 8-cube C standing at the back. At the level of its top row, glue 7 more cubes in front of that row so the whole top layer, seen from above, reads (from back to front): 3 squares, then 2 squares at the ends, then 3 squares, then 2 squares at the ends. That is the letter seen from the top.
3. From the front, the new cubes are hidden behind the C's top row, so the front view is still the C.

In shortYes. One way: the C at the back plus 7 cubes in its top layer reaching 3 cubes forward, 15 cubes in all. From the front it is still the C; from the top the layer shows the letter, 3 squares wide and 4 deep.

(iii) Now, can you glue even more cubes …

1. Add 3 more cubes in the second layer from the bottom, in a line going forward from the C's left column (under the left edge of the top layer).
2. From the front they hide behind the C's left column, and from the top they hide under the top layer, so the C and the top letter do not change.
3. Now look from the left side: the C's column is a full-height stroke at the back, the top layer is a bar along the top, and the 3 new cubes with the C's second cube make a middle bar. That is the letter F, 4 squares wide and 4 tall. (From the right side you would see the same F reversed.)

In shortYes. Adding 3 cubes to the solid of (ii), 18 cubes in all, gives the C from the front, the same letter from the top, and an F from the left side.

(iv) Can you think of other letter combinations …

1. Any three letters can work only if they fit together: the front and top views share the width, the front and side views share the height, and the top and side views share the depth.
2. Example: take a 3 by 3 by 3 block of 27 cubes and remove the centre cube and the 6 cubes at the centres of its faces. The 20 cubes left look like the letter O from the front, from the top and from the side.
3. Other combinations that work on a 3 by 3 grid include L-L-L and T-T-T (7 cubes each).

In shortYes, there are many; for example O from all three directions (a 3 by 3 by 3 block with its centre and face-centre cubes removed, 20 cubes), or L, L, L. There are several correct answers; this is one.

Watch this explained “Three planes, three views”, 5:51 into Front, top and side views, and what each one loses · हिंदी में देखें

Open NCERT p. 96Checked by computer

1. Read the three views as heights of stacks of cubes. The front view is 4 squares wide: the left half is 3 squares tall and the right half 2 squares tall. The side view shows heights 1, 1 and 3 from front to back. The top view is 3 rows deep: the back and middle rows are full, and the front row has squares only on the left half.
2. So the solid has: a front block on the left, 1 cube high; a middle row across the whole width, 1 cube high; and a back wall across the whole width whose left half is 3 cubes high and right half 2 cubes high. In each square of the top view the height is fixed by the front and side views, so the solid is completely determined.
3. Now compare with the pictures. (i) and (v) have a back wall with a flat top, so their front views would have no step. (vi) has a notch in the top of its back wall. (iii) has a front block at both ends with a slot between them, so its top view would have a front square on the right too. (iv) has only a narrow block on top at the back, so the right half of its front view is only 1 square high. (vii) has its tall part at the front, not the back.
4. Only (ii) has a front block on the left only, a full middle row, and a back wall that steps down from left to right.

AnswerOption (ii).

Watch this explained “Three planes, three views”, 5:51 into Front, top and side views, and what each one loses · हिंदी में देखें

Open NCERT p. 96One way to think about it

(i)-(iii)

1. Heights are read as stacks of cubes; 'back', 'middle', 'front' are the rows of the top view from top to bottom, and columns are counted from the left.
2. Top view (i): back row - columns 2 and 3; middle row - columns 1, 2, 3; front row - columns 1 and 3 (a gap in the middle).
3. Front view (ii): column heights 1, 2, 2. Side view (iii): heights 1, 1, 2 from front to back.
4. A height of 2 is allowed only where the front view and the side view both allow it: in the back row, in columns 2 and 3. Every other square gets 1 cube.

In shortPut 1 cube on each of the 7 squares of the top view, then 1 more cube on each of the 2 squares of the back row: 9 cubes. This is the only stack of cubes that gives all three views.

(iv)-(vi)

1. Top view (iv): back row 2 squares, middle row 2 squares, front row 1 square on the left.
2. Front view (v): heights 2, 2. Side view (vi): heights 1, 1, 2 from front to back, so a height of 2 is allowed only in the back row.

In shortPut 1 cube on each of the 5 squares of the top view and 1 more on each of the 2 back squares: 7 cubes. This is the only such stack.

(vii)-(ix)

1. Top view (vii): back row 2 squares, middle row 2 squares, front row 1 square on the left.
2. Front view (viii): heights 2, 3. Side view (ix): heights 1, 1, 3 from front to back, so the tall stacks must be in the back row: 2 cubes on the left and 3 on the right. The dashed line in the side view is the hidden top edge of the 2-cube stack behind the 3-cube one.

In shortPut 1 cube on each of the 5 squares of the top view, then make the back-left stack 2 cubes tall and the back-right stack 3 cubes tall: 8 cubes. This is the only such stack.

Watch this explained “Three planes, three views”, 5:51 into Front, top and side views, and what each one loses · हिंदी में देखें

Open NCERT p. 97Checked by computerReads two ways: both answers shown

1. The picture shows 10 cubes, in rows of 1, 2, 3 and 4 from the top, each row one cube lower than the row above.
2. A drawing of a solid can hide cubes behind the ones we see, so we must decide what holds the upper cubes up. The book gives no answer key.
3. Reading 1 (a real stack: every cube rests on cubes below it). We lead with this because the question calls it a stack. The top cube is 3 cubes above the floor, so 3 hidden cubes stand under it. Each cube in the second row needs 2 under it, and each cube in the third row needs 1. The bottom row sits on the floor.
4. Hidden cubes = 1 × 3 + 2 × 2 + 3 × 1 = 10. Total = 10 + 10 = 20.
5. Counted layer by layer from the top, this stack is 1 + 3 + 6 + 10 = 20 cubes, like a staircase built into a corner. All the hidden cubes sit behind the ones we see, so the picture looks exactly the same.
6. Reading 2 (count only the cubes the picture shows): 1 + 2 + 3 + 4 = 10. This can happen too: 10 cubes spread out on the floor in the right places look exactly like this picture from this one viewpoint.
7. That is the idea of this chapter: a flat drawing of a solid loses information, so one picture can fit more than one pile of cubes.

AnswerRead as a real stack, with hidden cubes holding the upper ones up: 20 cubes. Counting only the cubes you can see: 10 cubes.

Watch this explained “Two heaps, one drawing”, 6:38 into Drawing solids on an isometric grid

Open NCERT p. 97Checked by computer

1. A projection of a cube is its shadow on a plane when the light falls straight onto the plane. Its outline is always made from the shadows of the cube's edges, which come in 3 groups of 4 parallel edges.
2. If a face of the cube is parallel to the plane, the edges going towards the plane shrink to points and the shadow is a square.
3. If the cube is tilted about one edge only (one group of edges stays parallel to the plane), the shadow is a rectangle: one pair of sides is an edge length, the other is longer.
4. In any other position all 3 groups of edges show up in the outline, and the shadow is a hexagon with opposite sides equal and parallel. In general its sides have 3 different lengths.
5. When the cube is balanced on a corner with the opposite corner straight above it, all its edges cast shadows of the same length and the hexagon is regular - this is the isometric projection.

AnswerA square, a rectangle or a hexagon. The hexagon is regular only when the cube is balanced on a corner.

Watch this explained “What shapes can a shadow be?”, 0:48 into Isometric projection: the orientation that keeps all edges equal · हिंदी में देखें

Open NCERT p. 100One way to think about it

1. The 5 shapes of Fig. 4.8 all lie flat: the 4 cubes are in one layer. (As solids, the L and its mirror image are the same piece, because a cube piece can be turned over; the same goes for the S-shape.)
2. Gluing along faces does not force the cubes to stay in one layer. Start from 3 cubes in an L (a corner cube with one cube on each side of it).
3. Glue the 4th cube on top of the corner cube: a 'tripod' - one cube with three others on three of its faces, pointing in three different directions. This is new.
4. Glue the 4th cube on top of one of the end cubes of the L instead: the piece twists out of the layer like a screw. Its mirror image (the 4th cube on top of the other end cube) is also a solid piece, and no turning makes one into the other, just as a left glove cannot be turned into a right glove.
5. Checking every way of gluing 4 cubes face to face gives exactly these: 5 flat pieces and 3 that leave the layer.

In shortYes, 3 more: the tripod (a cube on top of the corner of an L of 3) and a left-handed and a right-handed twisted piece (a cube on top of an end of the L), so 8 ways in all. If a piece and its mirror image are counted as the same, the extra ones are 2 and the total is 7.

Watch this explained “Five was the flat answer”, 5:17 into Drawing solids on an isometric grid

Open NCERT p. 101One way to think about it

First figure (the L)

1. On isometric paper, vertical lines are the height, lines sloping down to the left are the depth, and lines sloping down to the right are the length. Draw edge by edge, counting grid units; for each edge decide whether it goes up or down, and draw it with or against that grid direction (the book's hint).
2. The book's figures are not drawn on a grid, so choose whole numbers that keep their shape. This L is 1 unit thick; its foot is about 3 units long and its upright about 3 units tall.
3. Draw the end of the foot as a 1 by 1 square face, then the foot's long side 3 units along the length direction, then the upright rising 3 units from the far end of the foot, 1 unit thick.

In shortAn L 1 unit thick, 3 units long along the ground and 3 units tall at its far end (1 unit by 3 by 3 overall). Other whole-number sizes with the same shape are also correct.

Second figure (the block on a base)

1. This figure is drawn with its front face flat to you and its depth slanting, so it has to be redrawn with the front face along the length direction of the grid.
2. It is a long, low base with an upright block standing on it; the upright is taller than it is wide, and both have the same depth. The drawing is not to scale, so choose whole-number sizes that keep this shape.
3. For example: base 6 units long, 2 deep and 2 high; upright block 2 units long, 2 deep and 3 tall, standing on the base 2 units from each end. Draw the base edge by edge, then the upright's edges rising from its top.

In shortFor example, a base 6 long, 2 deep and 2 high with an upright block 2 long, 2 deep and 3 tall standing on it, 2 units from each end. Other whole-number sizes that keep the same shape are also correct.

Third figure (the staircase)

1. This figure is also drawn with a flat front face, and its front face is 3 steps: each step is as high as it is long.
2. Take each step 1 unit high and 1 unit long, so the staircase is 3 units long and 3 units high. The book's slanted depth does not fix the depth in grid units; choose a depth, say 2 units.
3. Draw the front face as a staircase outline (up 3, across 1, down 1, across 1, down 1, across 1, down 1, back 3 along the ground), then draw every corner's depth edge 2 units along the depth direction and join the back ends.

In shortA staircase of 3 steps, each 1 unit high and 1 unit deep (3 units long and 3 high overall), drawn with a depth of 2 units. Other depths and sizes with the same step shape are also correct.

Watch this explained “Erase, or count”, 2:18 into Drawing solids on an isometric grid

Open NCERT p. 101One way to think about it

1. Follow the arrows from the red ball. The ball rolls along the front row of cube tops (level ground), turns, climbs up a step (the first up-arrow), crosses the top of that step, climbs up a second step (the second up-arrow), rolls along the right-hand row and then back along the back row, and arrives at the cube where it started.
2. Along the way it goes up two steps and never goes down a single step.
3. A real path cannot do that. If a path ends where it began, its height at the end equals its height at the start, so every climb must be matched by an equal drop. Here the path climbs twice and never drops, so it cannot close up in a real solid. The drawing only works because an isometric picture cannot show how far back something is: the back row that looks level with the start is really two steps higher.
4. To recreate it, use the hint: draw a part that can really be built, for example the front row and the two steps up, or the two steps and the back row. Mark the three directions (length, depth, height) on the grid and draw that part edge by edge. Joining the last part back to the start is only possible on paper.

In shortYes. The ball goes round a closed loop, climbing two steps and never going down, and returns to where it started - which is impossible in a real solid. Any single stretch of the path can be built from cubes; the whole loop exists only in the drawing.

Watch this explained “A route that only ever goes up”, 7:16 into Drawing solids on an isometric grid

Open NCERT p. 101One way to think about it

(i) Would it be possible to build a model out of actual cubes?

1. The triangle is three straight bars of cubes. Each bar runs along a different one of the three directions (length, depth, height), and each is 5 cubes long counting the cubes at the corners.
2. Any two bars joined at one right-angled corner can really be built.
3. But going round all three bars you move 4 steps along the length, 4 along the depth and 4 up (or down). These do not cancel, so the end of the third bar is not back at the start of the first: it is 4 steps further along each direction. It only looks joined because that shift points straight away from the viewer, which an isometric drawing cannot show. So a closed triangle cannot be built.
4. What can be built is the open version: three bars in a chain, each at a right angle to the last. Seen from exactly the right direction it looks like the drawing.
5. Its profiles: looking along any one direction, the bar that runs along that direction is seen end-on as a single square, and the other two bars appear at a right angle. So the front, top and side profiles are each an L-shape: two arms of 5 squares meeting at a corner square.

In shortNo, not as a closed triangle - going round it would need the three bars to cancel out, and they do not. An open chain of three bars can be built and looks the same from one viewpoint; its front, top and side profiles are each an L-shape of two 5-square arms meeting at a corner.

(ii) Recreate this on an isometric grid.

1. On isometric paper, draw one bar of 5 cubes along one grid direction, then from its end a second bar along the second direction, then a third bar along the third direction.
2. Draw each bar edge by edge, counting 5 units; the third bar ends exactly on the first cube on paper, because 4 steps along each of the three directions lands on the same point of the page.

In shortDraw three bars of 5 cubes, one along each of the three grid directions, each starting where the last ended; on paper the third bar meets the first.

(iii) Why does the illusion work?

1. An isometric drawing throws away one direction: the direction straight towards the viewer. Moving equal amounts along the length, depth and height moves along exactly that direction, so on paper it looks like no move at all.
2. Each corner of the drawing is a real, buildable corner, so the eye finds nothing wrong at any one place. The contradiction only appears when you go all the way round, and the picture has no way of showing that the end is really behind the start.

In shortBecause an isometric picture cannot show depth along the line of sight: 4 steps along each of the three directions lands on the same spot on paper, so the open end of the third bar is drawn on top of the start of the first, and every single corner looks correct.

Watch this explained “Three good corners, one bad closure”, 8:18 into Drawing solids on an isometric grid