Geometry and measures - Mensuration and calculation: know the formulae: circumference of a circle = 2πr = πd , area of a circle = πr² ; calculate: perimeters of 2D shapes, including circles; areas of circles and composite shapes; surface area and volume of spheres, pyramids, cones and composite solids

The idea

Know and apply formulae for circles: circumference C = 2πr = πd and area A = πr². For composite 2D shapes, add or subtract areas of standard shapes and sum only the external lengths for the perimeter. For 3D solids, formulae for volume (V) and surface area (SA) are required. Sphere: V=(4/3)πr³, SA=4πr². Pyramid: V=(1/3) × base area × height; its SA is the sum of the base area and the area of its triangular faces. Cone: V=(1/3)πr²h, total SA = πrl + πr² (curved area + base area). For composite solids, like a hemisphere on a cylinder, add the volumes. For surface area, add the external faces only (e.g. hemisphere's curved area, cylinder's curved area, and cylinder's circular base), excluding any internal join.

What earns the marks

• Correct substitution into a circle formula (C=2πr or A=πr²), distinguishing radius from diameter.
• Correctly calculating the volume or total surface area of a sphere, pyramid or cone.
• Decomposing a composite 2D or 3D shape into standard parts and calculating their individual measures.
• Combining measures for a composite solid, ensuring internal faces are correctly excluded from surface area calculations.

Common mistakes and how to put them right

• Using the diameter instead of the radius in a formula like πr² or (4/3)πr³.

• Forgetting to add the area of the circular base (πr²) when asked for the total surface area of a cone.

• Confusing a pyramid's or cone's slant height (l) with its perpendicular height (h) in formulae.

• Including the area of joined faces when calculating the surface area of a composite solid.

Exam tips

• Before substituting, explicitly write down the values for radius (r), diameter (d), height (h), and slant height (l) given in the question.
• For composite solids, sketch the individual components or list the external faces to ensure you are calculating the correct surface area.
• Use the π button on your calculator and only round the final answer to the required accuracy.