Chapter 14 — Math of Space: Surface Area and Volume

Chapter 14 is titled 'Math of Space: Surface Area and Volume'. It covers how to find the surface areas and volumes of cuboids, cylinders, cones, pyramids, and spheres, including hands-on experiments and guesstimate problems.

For a cuboid with length l, width w, and height h: TSA = 2(wl + hl + hw) and V = lwh. For a cube of side a, these simplify to TSA = 6a² and V = a³.

The curved surface area (CSA) of a cylinder with radius r and height h is 2πrh. If closed at both ends, the total surface area is 2πr(h + r); if closed at one end only, it is πr(2h + r).

A cone has exactly one-third the volume of a cylinder with the same base radius and height. So V (cone) = ⅓πr²h. The chapter verifies this experimentally by filling a cone with salt and emptying it into the matching cylinder — three full cones exactly fill it.

The slant height l is the distance from the apex to any point on the base circle. It is related to the height h and base radius r by the Baudhāyana–Pythagoras theorem: l² = h² + r².

For a sphere of radius r: surface area = 4πr² and volume = (4/3)πr³. Archimedes derived both these formulas around 225 BCE.

A hemisphere is formed when a sphere is cut into two equal halves by a plane through its centre. Curved surface area = 2πr², total surface area = 3πr², and volume = (2/3)πr³.

A pyramid's volume = ⅓ × area of base × height, regardless of whether the base is triangular, square, or any other flat shape. The chapter notes this is the same formula as for a cone, because a cone can be thought of as a pyramid with a circular base.

A cross-section is the shape obtained by taking a thin slice of a 3D object. For a cylinder sliced parallel to its base, the cross-section is a circle. The chapter uses this idea — stacking thin circular slices — to explain why the cylinder's volume formula V = πr²h is true.

Guesstimate problems ask for a reasonable estimate when exact data are unavailable. The chapter uses examples such as estimating how many golgappas can be served from a cylindrical vessel, and how many tennis balls fit in a classroom, applying the volume formulas for spheres and cylinders with stated assumptions.

Archimedes showed that the surface area of a sphere equals the curved surface area of a cylinder with the same radius r and height h = 2r that fits tightly over the sphere, giving 4πr².

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The chapter discusses packaging (a cube vs cuboid of the same volume needs less covering material), heat transfer, engineering, MRI cross-sections, map-making (cylindrical projection of the earth), and everyday objects like cylinders, cones, and hemispheres such as bowls, domes, and kaleidoscopes.