NCERT Solutions Ganita Prakash (Part 2) Chapter 4 –93Front view, top view and side view — Figure it Out

Book page 92 Updated on2026-09-05

Q1.

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

Answer

Yes. All three lines were chosen so that they have the same front view, and their front views are horizontal segments. For each line separately, the three view-lengths obey the Baudhayana Theorem.

Line	Front view	Top view	Side view
1	A horizontal segment	A horizontal segment of the same length	A point
2	The same horizontal segment	A slanting segment, a little longer	A short segment
3	The same horizontal segment	A more steeply slanting, longer segment	A longer segment

Take a segment whose ends differ by Δx across, Δy in depth and Δz in height. Then

front view f = √(Δx² + Δz²)
top view t = √(Δx² + Δy²)
side view s = √(Δy² + Δz²)
actual length l = √(Δx² + Δy² + Δz²)

Adding the first three squares counts each of Δx², Δy², Δz² exactly twice:

f² + t² + s² = 2 l²

All three lines in Fig. 4.6 lie in a horizontal plane, so Δz = 0 for each. Then f = |Δx| — the same for all three, which is why the front views agree — and s = |Δy|, so:

t² = Δx² + Δy² = f² + s²

So for these lines the top view is the hypotenuse of a right triangle whose legs are the front view and the side view. Line 1 has s = 0, so t = f; as the line is swung round in the horizontal plane, s grows and t grows with it, while f stays fixed.

Why no single view is enough: all three lines have identical front views but different actual lengths. Only when the side view is added can you tell them apart — and only all three together give l, through f² + t² + s² = 2l².

Q2.

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. If needed, see the next problem for clues.

Answer

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

Solid (orientation)	Front view	Top view	Side view
Cube, faces parallel to the planes	Square	Square	Square
Cuboid l × b × h, edges along the axes	Rectangle l × h	Rectangle l × b	Rectangle b × h
Parallelepiped, one face on the floor	Parallelogram	Parallelogram	Parallelogram
Cylinder standing upright	Rectangle 2r × h	Circle of radius r	Rectangle 2r × h
Cone standing on its base	Isosceles triangle, base 2r, height h	Circle of radius r	The same isosceles triangle
Triangular prism lying along the side direction	Triangle	Rectangle	Rectangle
Square pyramid standing on its base	Isosceles triangle	Square (with the diagonals showing the four slant edges)	Isosceles triangle

Why the front and side views of a cylinder and a cone are identical: both solids are symmetric about a vertical axis, so they look the same from every horizontal direction. Their front and side views must therefore agree. A cuboid has no such symmetry, so its three views are three different rectangles.

Tip: The three views are not independent. The height shown in the front view must match the height in the side view; the width in the front view must match the width in the top view; and the depth in the top view must match the depth in the side view. This is the standard check draughtsmen use.

Q3.

Match each of the following objects with its projections.

Answer

Take each object, look along the F, T and S arrows drawn beside it, and find the row of three outlines that fits. The arrows are not the same for every object, so the direction called ‘front’ changes from picture to picture.

Object	Front view	Top view	Side view
Measuring jug / mug	Body with a V-shaped spout notch at the rim	Circle with the spout and handle sticking out	Body with the handle in profile
Funnel	Downward triangle with a thin stem	Circle with a small circle (the stem) at the centre	The same triangle with stem
Hammer	Narrow head above a straight handle	Short bar — the head seen end on	The familiar hammer profile
Car	Bonnet, grille and headlamps	Roof and bonnet outline	Full side of the car with both wheels
Slide with a box on it	Tall narrow rectangle with the box as a small square	Long rectangle (the ramp) with the box inside it	Ladder and sloping ramp, with the box on the slope
Chair	Back and two front legs	Seat as a square with the back edge	Back, seat and legs in profile
Ceiling fan	Rod, motor and blades edge on	The three blades spread out	The same as the front view
Cooking pot / cooker	Body with the lid on top	Circle with the handle bar across it	Body with the handle sticking out

Why the fan is the easiest and the hammer the hardest: the fan has a clear axis, so its top view (three blades) is unmistakable and quite unlike its side view. The hammer looks like a plain bar from two directions and only shows its character from the third. Match the most distinctive view first, then confirm with the other two.

Tip: Two of the three views of a symmetric object are often identical — the fan and the funnel are examples. When two views agree, the object almost always has an axis of symmetry.

Was this helpful? Report an error