Solids [Surface Area & volume]
ICSE Class 9 · Mathematics · Chapter 20
What this chapter covers
- Solids, faces, edges and vertices
- Surface area of a cuboid
- Surface area of a cube
- Volume of a cuboid and a cube
- Open and closed cubes and cuboids
- Internal and external dimensions
- Volume as area of cross-section × height
- Problems involving cost
Key definitions
Total surface area is the sum of the areas of all the faces that bound a solid.
Capacity is the volume of liquid or other material that a hollow container can hold, and it is usually measured in litres.
External dimensions are the length, breadth and height of a hollow box measured across its outside surfaces.
Internal dimensions are the length, breadth and height of the empty space inside a hollow box, measured between its inner surfaces.
Key points
- A cuboid has 6 rectangular faces, 12 edges and 8 vertices; a cube is a cuboid with all edges equal.
- Total length of edges: cuboid 4(l + b + h), cube 12a.
- Total surface area: cuboid 2(lb + bh + hl), cube 6a2.
- Lateral surface area (area of four walls): cuboid 2(l + b)h, cube 4a2.
- Open at the top: cuboid 2(l + b)h + lb, cube 5a2; volume is the same whether the box is open or closed.
- Volume: cuboid lbh, cube a3; 1 litre = 1000 cm3 and 1 m3 = 1000 litres.
- Capacity uses internal dimensions; volume of material = external volume - internal volume.
- With wall thickness t: internal length l - 2t, breadth b - 2t; internal height h - 2t if closed, h - t if open at the top.
- For any solid of uniform cross-section, volume = area of cross-section × height (or length).
- In cost problems, a rate per m2 needs an area and a rate per m3 needs a volume; convert units before multiplying.
Questions answered
Define a cuboid and state how many faces, edges and vertices it has.
The sum of the lengths of all the edges of a cube is 96 cm. Find its total surface area and volume.
A cuboid measures 12 cm by 8 cm by 5 cm. Find its total surface area, lateral surface area and volume.
The volume of a cube is 343 cm3. Find its total surface area.
Express 2.4 m3 in litres, and explain why 1 m3 is not 100 cm3.
Find the area of sheet needed for a box without a lid measuring 25 cm by 20 cm by 10 cm.
A closed wooden box has external dimensions 24 cm by 18 cm by 12 cm, and the wood is 1.5 cm thick. Find its capacity and the volume of wood.
If the box in question 7 had no lid, what would its capacity be? Why does the answer change?
Past board exam questions
ICSE 2019 sample paper · 3 marks
Three cubes are kept adjacently, edge to edge. If the edge of each cube is 7 cm, find total surface area of the resulting cuboid.
Show answer
ICSE 2019 sample paper · 4 marks
The cross-section of a 6 m long piece of metal is a rectangle 8 cm wide and 6.5 cm high, with an isosceles triangle standing on its 8 cm top side whose two equal sides are 5 cm each. Calculate: (i) the area of the cross-section (ii) The volume of the piece of metal in cubic centimetres.