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NCERT Solutions Ganita Prakash (Part 2) Chapter 4 –774.2 Visualising Solids — Build it in Your Imagination

Book page 75 Updated on2026-09-05

Q1.
Picture your name, then read off the letters backwards. Make sure to do this by sight, not by sound — really see your name! Now try with your friend's name.
Answer

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

MEENA → A N E E M
RAGHAV → V A H G A R
SUBRAMANIAN → N A I N A M A R B U S

Notice the difference between the two ways of doing it. If you go by sound you have to say the name to yourself and then work backwards, which is slow and easy to lose track of. If you go by sight the letters are all there at once and you simply scan them the other way.

Why this exercise opens the chapter: everything that follows — nets, hidden faces, three views of a solid — asks you to hold a picture steady in your head and then turn it round or look at it from a new direction. This is that skill on a flat, easy object first.
Q2.
Cut off the four corners of an imaginary square, with each cut going between midpoints of adjacent edges. What shape is left over? How can you reassemble the four corners to make another square?
Answer

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

cut off 4 corners → a square, half the area
The four corner triangles are congruent right isosceles triangles; the inner square is the one that survives.

Why it is a square. Each cut joins midpoints of two adjacent sides, so all four cuts have the same length, and each corner cut removes a right isosceles triangle with legs a⁄2. At each vertex of the inner shape two 45° angles are removed from a 180° straight angle, leaving 90°.

Reassembling the four corners. The four triangles each have legs a⁄2 and hypotenuse a⁄√2. Put the four right angles together at one point; the legs match up in pairs and the four hypotenuses form the boundary. You get a square of side a⁄2.

Original area = a²
Inner square = a²⁄2 (half)
Four corners = 4 × ½ × (a⁄2) × (a⁄2) = a²⁄2 (the other half) ✓
Q3.
Mark the sides of an equilateral triangle into thirds. Cut off each corner of the triangle, as far as the marks. What shape do you get?
Answer

A regular hexagon.

Each side is divided into thirds; cutting the three corners leaves a hexagon with all six sides equal to one-third of a side.

Let each side be 3 units, so the marks are 1 unit apart. Each corner triangle has two sides of 1 unit with a 60° angle between them, so it is equilateral with side 1. Cutting all three corners leaves six sides, and each is 1 unit long: three of them are the middle thirds of the original sides, three are the cut edges.

Each interior angle of the hexagon = 180° − 60° = 120°
All six sides = 1 unit ⇒ regular hexagon
Area left = 1 − 3 × (1⁄9) = 2⁄3 of the triangle
Q4.
Mark the sides of a square into thirds and cut off each of its corners as far as the marks. What shape is left?
Answer

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.

The 8 sides go: 1, √2, 1, √2, 1, √2, 1, √2
1 ≠ √2 ≈ 1.414, so the sides are not all equal
All 8 angles are equal (135° each)
Area left = 9 − 4 × ½ × 1 × 1 = 9 − 2 = 7 sq. units, i.e. 7⁄9 of the square
Why the triangle gives a regular figure and the square does not: in the triangle the corner angle is 60°, so the corner piece cut off is equilateral and its third side is also 1. In the square the corner angle is 90°, so the piece cut off is right-angled and its hypotenuse is √2 — longer than the sides it replaced. To make the octagon regular you would have to cut at a different distance, not at the thirds.
Try This: Where should you mark a square of side 3 so that the octagon comes out regular? You need the cut length x to satisfy x√2 = 3 − 2x, which gives x = 3⁄(2 + √2) ≈ 0.879 — not one-third.
Q5.
A solid whose profile has a square outline
Answer

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

SolidViewpoint that gives a square
CubeAny direction perpendicular to a face
Cuboid with a square cross-sectionLooking along its length
Cylinder whose height equals its diameterLooking at it from the side
Square pyramidLooking straight down from above
Why so many answers: the profile only records the outline of the shadow. Everything about the depth of the solid is thrown away, so completely different solids can leave the same hole in the wall.
Q6.
A solid whose profile has a circular outline
Answer

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.
Did you know? The sphere is the only solid whose profile is a circle from every direction. That is why a ball rolls equally well whichever way you push it.
Q7.
A solid whose profile has a triangular outline
Answer

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.
Q8.
A solid with a rectangular profile from one viewpoint and a circular profile from another viewpoint
Answer

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

cylinder from the side from the top
One solid, two very different profiles — the whole point of taking more than one view.
Side view: rectangle, height h by diameter 2r
Top view: circle of radius r
Q9.
A solid with a circular profile from one viewpoint and a triangular one from another viewpoint
Answer

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

From directly above (along the axis): a circle of radius r
From the side (perpendicular to the axis): an isosceles triangle of base 2r and height h

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.

Q10.
A solid with a rectangular profile from one viewpoint and a triangular one from another viewpoint
Answer

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

Along its length: the triangular end face
From the side: a rectangle, length by height

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

Q11.
A solid with a trapezium shaped profile from one viewpoint and a circular one from another viewpoint
Answer

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.

From above: a circle (the wider rim)
From the side: a trapezium — the two circular rims give the parallel sides, the slanting wall gives the other two
Why a plain cone will not do: the cone comes to a point, so its side profile is a triangle, not a trapezium. You need the top to be cut off flat so that the profile has two parallel sides.
Q12.
A solid with a pentagonal profile from one viewpoint and a rectangular one from another viewpoint
Answer

A pentagonal prism — the shape of many pencils and of a five-sided pillar.

Along its length: the regular pentagon end face
From the side: a rectangle, length by the width of the pentagon

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.

Q13.
Are there unique solids for each of the conditions, or can you come up with multiple possibilities?
Answer

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

ConditionOne answerAnother answer
Square profileCubeSquare pyramid seen from above
Circular profileSphereCylinder seen end on
Rectangle + circleCylinderSphere squashed into an oval (a lemon)
Circle + triangleConeA spinning top
Trapezium + circleFrustum of a coneA bucket, a flower pot
Why uniqueness fails: a profile is a shadow. It records the outline only — nothing about what is behind it, nothing about hollows or dents, nothing about the depth. Two solids whose outlines happen to agree from one direction are indistinguishable in that view. This is exactly the reason engineers never draw one view of a machine part: they draw three. Even then, as Fig. 4.6 shows, three views can still be shared by different objects.
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