Std 5 Mathematics · Chapter 13

13. Three Dimensional Objects and Nets

Solutions to Problem Set 51 and the chapter activities, with diagrams — views of 3D objects, nets, pentominoes and the dice riddle. Maharashtra State Board (Balbharati), Grade Five Mathematics, Part Two.

Key Ideas

Three dimensional objects have length, breadth and height (a box, a book, a glass). Their pictures on paper are two dimensional.

Views: the same object looks different from the front, from a side and from above. The view from above is also called the top view.

Net: a flat shape that can be folded to make a 3D object. A cube has 6 equal square faces; a cuboid has 6 rectangular faces.

Pentomino: five equal squares joined side to side. Some pentominoes fold into an open box (a cube without a lid).

Problem Set 51

1The first column shows a structure made of blocks. The other columns show different views of the structure in two dimensions. Say whether each view is from the front, from a side or from above.

(1) Three cubes in a row

1st view (3 squares in a row): from the front
2nd view (1 square): from a side
3rd view (3 squares in a row): from above

The front view and the top view look the same because the structure is only one cube deep and one cube high.

(2) Three cubes in a row with one cube on the middle

1st view (3 squares in a row): from above
2nd view (2 squares, one on the other): from a side
3rd view (T-shape): from the front

From above, the top cube hides the middle cube below it, so only 3 squares are seen.

(3) Four cubes (two in front, two behind) with one cube on top of the back-left cube

1st view (2 squares with one on the right): from the (right) side
2nd view (2 squares with one on the left): from the front
3rd view (2 × 2 square): from above

From the front, the raised cube is on the left. From the right side, the back row is on the right, so the raised cube appears on the right.

2Draw three pictures of each of these three dimensional objects — a table, a chair and a water-bottle — as viewed from the front, from a side and from above.

Table — From the front

Table — From a side

Table — From above

Chair — From the front

Chair — From a side

Chair — From above

Water-bottle — From the front

Water-bottle — From a side

Water-bottle — From above

Table: from the front and side we see the top and two legs (the side view is narrower); from above we see only the rectangular top.
Chair: from the front we see the back-rest, seat and legs; from the side it looks like the letter h (back-rest, seat and legs); from above we see the seat, with the thin top edge of the back-rest behind it.
Water-bottle: it looks the same from the front and the side; from above we see a circle (the base) with a small circle (the mouth) at the centre.

These are simple sample sketches. Draw the objects in your own home or classroom; your drawings will depend on their actual shapes.

Nets

AActivity: Draw the nets on card sheet, cut them out and find the shapes of the boxes they form.

Net (1): six equal squares

Net (2): four long rectangles and two small end pieces

Net (1) has six equal squares. When folded, it forms a cube.
Net (2) has four equal rectangles in a strip and two smaller pieces at the ends. When folded, it forms a cuboid (a long box).

BActivity: Make open boxes from the five-square nets given. Try to find out other five-square nets that can be used to make open boxes.

There are 12 different pentominoes, and exactly 8 of them fold into an open box. The textbook shows four of them (the one beside the text and the three in the activity). The other four are:

X (plus shape)

In each of these, one square becomes the base and the other four fold up to make the four walls. The remaining four pentominoes (a straight line of 5, the U-shape, the P-shape and the V-shape) cannot make an open box, because two of their squares would land on the same side.

★A riddle: the net of a cube-shaped dice is given. If a dice is made of this net, which of the shapes (1) to (5) will it definitely not resemble?

The net, with the number of dots on each face

Opposite faces: in the row of four (1, 2, 6, 5), faces one apart are opposite: 1 is opposite 6 and 2 is opposite 5. The two remaining faces, 3 and 4, are opposite each other.

Rule: two opposite faces can never be seen together.

Check (3): it shows 6 on top, 2 on the left and 5 on the right. But 2 and 5 are opposite faces, so they cannot both be seen at once.

The dice will definitely not resemble shape (3).

Shapes (1), (2), (4) and (5) show 6-3-2, 1-2-3, 4-2-1 and 6-2-4. In each, no two visible faces are opposite. When the net is folded, the dice can be turned to look exactly like each of them.

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