Surface Areas and Volumes

A cuboid is a solid bounded by six rectangular faces meeting at right angles. It is described by three measurements: length (l), breadth (b) and height (h). A cube is a special cuboid in which all edges are equal, l = b = h = a.

Every cuboid has three pairs of identical opposite faces, so its total surface area (TSA) is the sum of the areas of all six faces: two of size l × b, two of size b × h and two of size h × l. This gives TSA = 2(lb + bh + hl). If we only whitewash or paint the four vertical (side) walls — not the top and bottom — we use the lateral surface area (LSA), the perimeter of the base times the height: LSA = 2h(l + b).

The volume of a cuboid is the space it occupies: V = l × b × h. The diagonal (the longest segment inside the cuboid, joining opposite corners) is d = √(l² + b² + h²).

For a cube of edge a, these formulas simplify since l = b = h = a: TSA = 6a², LSA = 4a², V = a³ and diagonal d = a√3.

Worked Example: A room is 5 m long, 4 m wide and 3 m high. Find the area of its four walls and ceiling, and the cost of whitewashing at Rs 7.50 per m².

• Area of four walls (LSA) = 2h(l + b) = 2 × 3 × (5 + 4) = 54 m²
• Area of ceiling = l × b = 5 × 4 = 20 m²
• Total area to whitewash = 54 + 20 = 74 m²
• Cost = 74 × Rs 7.50 = Rs 555

A right circular cylinder has two congruent circular bases of radius r, placed directly one above the other at a perpendicular distance h, the height. Think of a pipe, a battery, or a drinking glass.

Unrolling the curved (lateral) surface flat gives a rectangle of length equal to the circumference of the base (2πr) and breadth equal to h. So the curved surface area (CSA) is CSA = 2πrh. The total surface area (TSA) adds the two circular ends: TSA = 2πrh + 2πr² = 2πr(r + h).

The volume of a cylinder equals the area of its circular base multiplied by its height: V = πr²h.

Worked Example: Find the curved surface area, total surface area and volume of a right circular cylinder of radius 7 cm and height 10 cm. (Take π = 22/7.)

• CSA = 2πrh = 2 × 22/7 × 7 × 10 = 440 cm²
• TSA = 2πr(r + h) = 2 × 22/7 × 7 × (7 + 10) = 748 cm²
• Volume = πr²h = 22/7 × 49 × 10 = 1540 cm³

A right circular cone has a circular base of radius r and a single apex directly above the centre of the base, at perpendicular height h. The slant height l is the distance from the apex to the edge of the base along the curved surface. Since r, h and l form a right-angled triangle inside the cone, l = √(r² + h²).

Unrolling the curved surface gives a sector of a circle of radius l. This curved surface area (CSA) works out to CSA = πrl. Adding the flat circular base gives the total surface area (TSA): TSA = πrl + πr² = πr(l + r).

The volume of a cone is exactly one-third the volume of a cylinder with the same base and height: V = (1/3)πr²h.

Worked Example: A conical tent has base radius 7 m and height 24 m. Find its slant height, the curved surface area of canvas needed, and the volume of air inside it. (Take π = 22/7.)

• Slant height l = √(r² + h²) = √(49 + 576) = √625 = 25 m
• CSA = πrl = 22/7 × 7 × 25 = 550 m²
• Volume = (1/3)πr²h = 1/3 × 22/7 × 49 × 24 = 1232 m³

A sphere (like a ball) of radius r has curved surface area SA = 4πr² and volume V = (4/3)πr³. A sphere has only one curved surface — there is no separate total surface area for a full sphere.

A hemisphere is obtained by cutting a sphere through its centre into two equal halves, giving a curved surface plus one flat circular face. Its curved surface area is exactly half that of the full sphere: CSA = 2πr². Its total surface area adds the flat circular face: TSA = 2πr² + πr² = 3πr². Since a hemisphere is exactly half of a sphere, its volume is V = (2/3)πr³.

Worked Example: Find the surface area and volume of a sphere of radius 7 cm, and the CSA, TSA and volume of a hemisphere of the same radius. (Take π = 22/7.)

• Sphere surface area = 4πr² = 4 × 22/7 × 49 = 616 cm²
• Sphere volume = (4/3)πr³ = 4/3 × 22/7 × 343 ≈ 1437.33 cm³
• Hemisphere CSA = 2πr² = 2 × 22/7 × 49 = 308 cm²
• Hemisphere TSA = 3πr² = 3 × 22/7 × 49 = 462 cm²
• Hemisphere volume = (2/3)πr³ = 2/3 × 22/7 × 343 ≈ 718.67 cm³

Many real objects combine the basic solids studied in this chapter — a capsule is a cylinder with two hemispherical ends, a wooden toy may be a cone mounted on a hemisphere, a tent may be a cylinder topped by a cone. To find the outer surface area of such a composite solid, add only the surfaces visible from outside; wherever two solids are joined, the flat circular face at the joint is hidden and must not be counted.

To find the volume of a composite solid, simply add the volumes of the separate parts, since the total space occupied is the sum of the spaces occupied by each piece.

Two other ideas commonly tested alongside surface area and volume are:

• Unit conversion: 1 m = 100 cm, so 1 m³ = 100 × 100 × 100 = 10,00,000 cm³, and 1 litre = 1000 cm³.
• Melting and recasting: when a solid is melted and recast into another shape, its volume stays the same even though its surface area changes.

Worked Example: A wooden article was made by scooping out a hemisphere of radius 3.5 cm from each end of a solid cylinder of the same radius and height 10 cm. Find the total surface area of the article. (Take π = 22/7.)

• The curved surface of the cylinder stays fully visible, and each flat end is replaced by a curved hemispherical surface of the same radius.
• Total surface area = CSA of cylinder + 2 × CSA of hemisphere = 2πrh + 2(2πr²) = 2πr(h + 2r)
• = 2 × 22/7 × 3.5 × (10 + 7) = 2 × 11 × 17 = 374 cm²