IP EMaths Notes (Upper Sec, Year 3-4): 11) Mensuration and 3D Geometry

Build a formula bank for quick recall and keep units consistent. Draw nets for surface-area questions and label every edge before calculating.

Formula essentials

• Area of circle: \(A = \pi r^{2}\), circumference: \(C = 2\pi r\).
• Curved surface area of a cylinder: \(2\pi rh\); total surface adds two bases \(2\pi r^{2}\).
• Volume of a prism: base area \(\times\) perpendicular height.
• Volume of a cone: \(V = \tfrac{1}{3}\pi r^{2}h\).

Worked example - Composite solid volume

A solid consists of a cylinder of radius \(\pu{3.0 cm}\) and height \(\pu{8.0 cm}\) with a hemisphere of the same radius attached on top. Find the volume and give the answer to \(3\) significant figures.

1. Cylinder volume: \(V_\text{cyl} = \pi r^{2}h = \pi (3.0)^{2}(8.0) = 72\pi \approx \pu{226 cm3}\).
2. Hemisphere volume: half of a sphere: \(V_\text{hem} = \tfrac{2}{3}\pi r^{3} = \tfrac{2}{3}\pi (3.0)^{3} = 18\pi \approx \pu{56.5 cm3}\).
3. Total: \(V \approx (72\pi + 18\pi) = 90\pi \approx \pu{283 cm3}\) to 3 s.f.

Try this

A frustum is formed by cutting the top \(\pu{4 cm}\) of a right cone whose base radius is \(\pu{6 cm}\) and height \(\pu{15 cm}\). Find the volume of the frustum, giving your answer in \(\pu{cm3}\).