Mensuration
Perimeter, area, surface area and volume of 2D and 3D shapes, including circles, prisms, cylinders, cones and spheres.
Learn
Mensuration, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Calculate the circumference and area of a circle
Circumference = πd (or 2πr); area = πr². The classic slip is using the diameter in the area formula - halve it first. "In terms of π" means leave π in the answer: a circle of radius 5 has area 25π, exactly.
Find arc lengths and sector areas
A sector is a fraction of the circle, set by its angle. Arc length = × πd and sector area = × πr² - a 60° sector is exactly one sixth of its circle. For perimeter of a sector, add the two radii to the arc.
Find the volume and surface area of prisms and cylinders
A prism has the same cross-section all the way through: volume = area of cross-section × length. A cylinder is a circular prism: V = πr²h, and its surface unrolls into two circles plus a rectangle, giving 2πr² + 2πrh.
Find the volume and surface area of pyramids, cones and spheres
Pyramid: V = × base area × height. Cone: V = πr²h, with curved surface πrl (l is the slant height). Sphere: V = πr³, surface 4πr². These formulae are on the exam formulae sheet - the skill is choosing and combining them.
Convert between units of area and volume - cm² to m², cm³ to m³
Lengths convert by 100, but areas convert by 100² = 10 000 and volumes by 100³ = 1 000 000: 1 m² = 10 000 cm². Convert the UNITS' factor, not just the number - or convert lengths before you multiply, which avoids the issue entirely.
Break compound solids into simpler shapes
A silo is a cylinder plus a hemisphere; a shed is a cuboid plus a triangular prism. Find each piece's volume and add - or subtract for holes. For surface area, count only faces on the OUTSIDE; the joining faces disappear.
Practice
Try a Mensuration question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
A circle has radius 6 cm. Work out its area. Use the π button and give your answer to 1 decimal place.
- 37.7 cm²
- 36.0 cm²
- 18.8 cm²
- 113.1 cm²
Show the answer and the working
Answer: 113.1 cm²
The area of a circle is π × radius².
- Area of a circle = π × radius².
- Square the radius: 6² = 36.
- Multiply by π: 36 × π ≈ 113.10.
- Rounded to 1 d.p. the area is 113.1 cm².