Surface area and volume
Section: Mensuration | Syllabus: Cambridge IGCSE Mathematics (0580)
Volume and Surface Area of Cuboids and Prisms
A cuboid is a 3D shape with six rectangular faces. A prism is any solid with a constant cross-section along its length - a cuboid is simply a prism with a rectangular cross-section.
- Volume of a cuboid =length
- Volume of a prism =cross-sectional area
- Surface area of a cuboid = sum of the areas of all 6 faces (3 pairs of equal rectangles)
- Surface area of a prism = sum of the areas of all its faces (the two end cross-sections plus the rectangular side faces)
Worked Example: Finding the Surface Area of a Cuboid
- Question: A cuboid has length 8 cm, width 5 cm and height 3 cm. Find its surface area.
- Step 1: Three pairs of faces: 85,\ 83,\ 53
- Step 2: Areas: 40, 24, 15
- Step 3: 2×(40+24+15)=279=158
- Answer: 158 cm²
Worked Example: Finding the Volume of a Triangular Prism
- Question: A prism has a triangular cross-section with base 6 cm and height 4 cm, and the prism itself is 10 cm long. Find its volume.
- Step 1: Cross-sectional area: (1/2)64=12
- Step 2: Volume: 1210=120
- Answer: 120 cm³
Common Mistakes
MistakeForgetting one of the three pairs of faces when finding a cuboid's surface area, or double-counting a pair
Fixsystematically identify all 3 distinct rectangle sizes first, then double each before adding
MistakeUsing the prism's length in place of the cross-sectional area, or vice versa, when finding volume
Fixalways find the cross-sectional area first (using the correct 2D shape formula), then multiply by the prism's length
Volume and Surface Area of Cylinders
A cylinder is a prism with a circular cross-section. Its volume and surface area combine circle formulas with the cylinder's height.
- Volume of a cylinder =π r^2h
- Curved surface area of a cylinder =2π rh
- Total surface area of a cylinder =2π rh+2π r^2 (curved surface plus the two circular ends)
Worked Example: Finding the Volume of a Cylinder
- Question: A cylinder has a radius of 6 cm and a height of 15 cm. Find its volume, correct to 3 significant figures.
- Step 1: π6^215
- Step 2: =π3615
- Step 3: 1696.46
- Answer: 1700 cm³ (3 s.f.)
Common Mistakes
MistakeForgetting to add the two circular ends when finding total surface area, giving only the curved surface area
Fixtotal surface area needs both the curved part (2πrh) and the two circles (2πr²) added together
MistakeUsing the diameter instead of the radius in the volume formula πr²h
Fixalways check whether the given measurement is a radius or a diameter, and halve a diameter before substituting
Volume and Surface Area of Spheres
A sphere is a perfectly round 3D shape, and a hemisphere is exactly half a sphere - both use formulas based on the radius.
- Volume of a sphere =(4/3)π r^3
- Surface area of a sphere =4π r^2
- A hemisphere's curved surface area is half a sphere's: 2π r^2; its total surface area also includes the flat circular face: 2π r^2+π r^2=3π r^2
Worked Example: Finding the Volume of a Sphere
- Question: A spherical ball has a radius of 9 cm. Find its volume, correct to 3 significant figures.
- Step 1: (4/3)×π9^3
- Step 2: 9^3=729
- Step 3: (4/3)×π7293053.63
- Answer: 3050 cm³ (3 s.f.)
Common Mistakes
MistakeForgetting the 4/3 fraction in the sphere volume formula, or misremembering it as 3/4
Fixthe sphere volume formula is always (4/3)πr³ - it can be checked against the formula sheet if unsure
MistakeUsing the total surface area formula for a hemisphere (3πr²) when only the curved part is needed, or vice versa
Fixcheck whether the question wants just the curved (domed) surface, or the total including the flat circular face
Volume and Surface Area of Cones
A cone narrows from a circular base to a single point (the apex). Its formulas use the radius, the vertical height, and the slant height (the distance from the apex to the edge of the base).
- Volume of a cone =(1/3)π r^2h, where h is the vertical (perpendicular) height
- Curved surface area of a cone =π rl, where l is the slant height (not the vertical height)
- Total surface area of a cone =π rl+π r^2 (curved surface plus the circular base)
Worked Example: Finding the Volume of a Cone
- Question: A cone has a radius of 5 cm and a height of 12 cm. Find its volume, correct to 3 significant figures.
- Step 1: (1/3)×π5^212
- Step 2: =(1/3)×π2512=100π
- Step 3: 314.16
- Answer: 314 cm³ (3 s.f.)
Common Mistakes
MistakeUsing the vertical height in the curved surface area formula, instead of the slant height
Fixthe curved surface area formula πrl always needs the slant height l, not the perpendicular height h - these are usually different values
MistakeForgetting the 1/3 fraction in the cone volume formula
Fixa cone's volume is always exactly a third of the cylinder with the same base and height - the 1/3 must not be dropped
Volume of Pyramids
A pyramid narrows from a polygon base to a single apex point. Its volume uses the same one-third idea as a cone, but with the base area instead of a circle's area.
- Volume of a pyramid =(1/3) area, where height is the perpendicular distance from the base to the apex
- The base can be any polygon - the base area must be calculated first using the correct 2D formula for that shape
- This formula works for any pyramid, whether the base is a square, rectangle, triangle or any other polygon
Worked Example: Finding the Volume of a Pyramid
- Question: A pyramid has a rectangular base measuring 6 cm by 5 cm, and a height of 9 cm. Find its volume.
- Step 1: Base area: 65=30
- Step 2: Volume: (1/3)309=90
- Answer: 90 cm³
Common Mistakes
MistakeMultiplying the base's side lengths directly into the volume formula without first treating them as an area
Fixalways calculate the 2D base area as a separate first step, using the correct formula for that shape
MistakeUsing a slanted edge length as the height, instead of the perpendicular height from the base to the apex
Fixthe height in the pyramid formula must be measured straight up (perpendicular) from the base plane to the apex, not along a sloping edge
Combining Volume and Surface Area Formulas
Extended Only
Some problems require setting up an equation by combining the volume or surface area formulas of two different solids, then solving algebraically for an unknown.
- Write an expression for each solid's volume or surface area separately, using the given information
- Set the two expressions equal to each other (or use the relationship given in the question) to form an equation
- Simplify the equation - dividing through by any common factor (such as π) often makes it much easier to solve
Interactive revision notes, videos and practice questions load below.