Figure it Out — Projections and Views — Exploring Some Geometric Themes

1. 13 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 92

   Observe the front view, top view and side view of the different lines in Fig. 4.6. Is there any relation between their lengths?

   Hint. Set up the line as the diagonal of a box whose edges run along the three axes.

   Set the situation up in coordinates. Take a line segment in space and let its ends differ by a along the length axis, b along the depth axis and c along the height axis. The segment is then the long diagonal of a box with edges a, b and c, and its true length is l = √(a² + b² + c²) (the theorem applied twice).

   Now read off the three views. Each view flattens one of the three directions: · Front view (on the vertical plane) keeps length and height, loses depth → length √(a² + c²) · Top view (on the horizontal plane) keeps length and depth, loses height → length √(a² + b²) · Side view (on the side plane) keeps depth and height, loses length → length √(b² + c²)

   The relation. Add the squares of the three view-lengths: (a² + c²) + (a² + b²) + (b² + c²) = 2(a² + b² + c²) = 2l²

   So for any segment,

   (front view)² + (top view)² + (side view)² = 2 × (true length)²

   A worked check. Take a = 3, b = 4, c = 12. Then l² = 9 + 16 + 144 = 169, so l = 13. Front = √(9 + 144) = √153, top = √(9 + 16) = 5, side = √(16 + 144) = √160. Squares: 153 + 25 + 160 = 338 = 2 × 169 ✓

   Two things this makes obvious. · Every view is at most as long as the segment itself, since each drops one positive term. A projection can shorten a line but never lengthen it. · A view equals the true length exactly when the segment is parallel to that plane — for instance the front view equals l when b = 0, meaning the segment has no depth component at all.

   ✦ Yes. If the segment differs by a, b, c along the three axes, the three views have lengths √(a²+c²), √(a²+b²) and √(b²+c²), so the sum of their squares is twice the square of the true length. Each view is never longer than the segment, and equals it exactly when the segment is parallel to that plane.

2. 25 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 92

   Find the front view, top view and side view of each of the following solids, fixing its orientation with respect to the vertical, horizontal and side planes: cube, cuboid, parallelepiped, cylinder, cone, prism and pyramid.

   Hint. Choose the simplest orientation — faces parallel to the planes — and then flatten one direction at a time.

   Fix the orientation first, since the answer depends on it. Take each solid sitting squarely: its natural base on the horizontal plane, and its front face parallel to the vertical plane. Then each view is found by flattening one direction.

   Solid	Front view	Top view	Side view
   Cube (side a)	square a × a	square a × a	square a × a
   Cuboid (l × b × h)	rectangle l × h	rectangle l × b	rectangle b × h
   Parallelepiped (slanted)	parallelogram	parallelogram	parallelogram
   Cylinder (radius r, height h)	rectangle 2r × h	circle of radius r	rectangle 2r × h
   Cone (radius r, height h)	triangle, base 2r, height h	circle of radius r, with the apex projecting to the centre	triangle, base 2r, height h
   Triangular prism (lying with triangular ends facing sideways)	rectangle	rectangle	triangle
   Square pyramid (base on the ground)	triangle	square with its two diagonals	triangle

   Two entries worth explaining.

   The pyramid's top view. Looking straight down, the square base gives a square outline, and the apex projects to the centre of that square. The four slanted edges run from the apex to the four corners, so they appear as the two diagonals of the square. The outline alone would not distinguish a pyramid from a flat square — the diagonals are the information that identifies it.

   The cone's top view. The same idea: the outline is a circle and the apex lands at its centre. Since the curved surface has no edges, no lines appear inside — just the circle with a marked centre.

   The general rule to carry away. A view shows the outline of the solid from that direction, plus any edges that are visible. Solids that are round in one direction give a circular or straight-edged view depending on which direction is flattened — a cylinder is a rectangle from two directions and a circle from the third.

   ✦ Cube: three squares. Cuboid: three rectangles, l×h, l×b, b×h. Parallelepiped: three parallelograms. Cylinder: rectangle, circle, rectangle. Cone: triangle, circle, triangle. Triangular prism: rectangle, rectangle, triangle. Square pyramid: triangle, square with diagonals, triangle.

3. 33 marksGanita Prakash Cl-8 Part 2, Figure it Out, page 93

   Match each of the given objects with its front, top and side projections.

   Hint. Match on the distinguishing feature, not the outline — several objects share an outline.

   The objects and their projections are printed diagrams, so they are not reproduced here. What follows is the matching method, which is what the question tests.

   Step 1 — start with the view that is most distinctive. In most sets one view separates the objects sharply. A circle in some view means the object is round in that direction — a cylinder, cone or sphere. A triangle means it tapers. Use that view to split the objects into groups, then use a second view to separate within a group.

   Step 2 — check the dimensions agree across views. Adjacent views must share a measurement, and this is the strongest test available: · front and top views must have the same width · front and side views must have the same height · top and side views must have the same depth Any proposed match where these do not agree is wrong, whatever the shapes look like.

   Step 3 — look at internal lines, not just outlines. The outline alone rarely decides it. A square with its diagonals is a pyramid from above; a plain square is a cube or a cuboid. A circle with a dot at the centre is a cone from above; a plain circle is a cylinder or a sphere. The lines inside the view carry the information the outline throws away.

   Step 4 — check for solids that share a view. A cylinder and a cuboid can both give a rectangle; a cone and a triangular prism can both give a triangle. That is exactly why three views are provided — no single one identifies the solid, but the three together usually do.

   Step 5 — verify by rebuilding. Once you have a match, work the other way: imagine the solid and check that all three of its views come out as printed. A match that survives being checked backwards is almost always right.

   ✦ Match on the most distinctive view first (a circle means round, a triangle means tapering), then confirm that adjacent views share the width, height and depth they must share, and use the lines inside each view — such as the diagonals that mark a pyramid — to separate solids with the same outline.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 8 Mathematics textbook Ganita Prakash Part 2, Reprint 2026-27 (hegp204.pdf), where this is Chapter 4 (pages 70-102) — the eleventh chapter of the Class 8 course and the longest in the book. Like the rest of Part 2 it carries NO printed answer key, so every formula and count was derived and then independently recomputed in Python before being written: the Sierpinski Carpet recurrences (R_n = 8^n, H_n = (8^n - 1)/7), the Sierpinski Triangle counts (3^n and (3^n - 1)/2), the areas (8/9)^n and (3/4)^n, the Koch side count 3 x 4^n and perimeter 3 x (4/3)^n, and the face/edge/vertex formulas for prisms and pyramids (checked against Euler's relation for every case). THIS IS A HEAVILY VISUAL CHAPTER, so figure-only items are handled in one of two ways and never guessed. (1) MEASURED FROM THE RENDERED PAGE: the cube-stack count on page 97 was settled by rendering the figure at 400 dpi and observing that each row sits one step BACK as well as one step up (every bottom cube shows its full top face), which makes it a square-layered step pyramid of 16 + 9 + 4 + 1 = 30 cubes rather than the ten visible; and the three letters in the page-96 puzzle were read off the printed pixel glyphs at 700-900 dpi as C (front), A (top) and F (side). (2) FLAGGED AND ANSWERED BY METHOD: the six candidate cube nets, the projection-matching sets, the cube-combination views, the isometric figures to copy, the rolling ball and the impossible triangle are all printed diagrams; each solution gives the full method and reasoning and says plainly that the diagram is not reproduced. ONE ITEM IS LEFT OPEN BY THE BOOK ITSELF and is reported as such: the 30 x 12 x 12 shortest-path Try This on page 87, where the book computes 42 cm and 40 cm for two unfoldings (24^2 + 32^2 = 1600 verified) and then says all unfoldings must be listed to find the answer — so the solution establishes only that the shortest path is at most 40 cm. The tetracube count in the page-100 exercise was verified by exhaustive computer enumeration: 8 arrangements up to rotation, 7 up to rotation and reflection, of which 5 are flat.. Questions are referenced from the NCERT textbook for identification.