Volume of solids
Volume measures the space inside a 3-D object, in cubic units (cm³, m³). For any prism, volume = area of cross-section × length.
Key idea
Cuboid: V = l w h | Cylinder: V = πr²h | Cone: V = ⅓πr²h | Sphere: V = 4/3 πr³
Surface area
Surface area is the total area of all faces or curved surfaces, in square units. For a cuboid, SA = 2(lw + lh + wh). For a sphere, SA = 4πr². The curved surface of a cylinder is 2πrh, so the total is 2πr² + 2πrh.
Worked example
A cylinder has radius 7 cm and height 10 cm (π = 22/7). Volume = πr²h = 22/7 × 49 × 10 = 1540 cm³. Curved surface area = 2πrh = 2 × 22/7 × 7 × 10 = 440 cm².
Remember
- Volume uses cubic units; surface area uses square units.
- A cone's volume is exactly one third of the cylinder with the same base and height.
Choosing the right formula
First identify the solid, then decide whether the question asks for volume (the space filled) or surface area (the material covering it). A common exam task combines shapes: for example a solid made of a hemisphere sitting on top of a cylinder needs the two separate volumes added together. Always substitute the radius, not the diameter, and finish with the correct cubic or square unit.