Surface area and volume of 3D solidsIB MYP Maths Extended: Revision notes

Volume is the space inside a solid, in cubic units such as cm$^3$. Surface area is the total area of all the faces, in square units such as cm$^2$. When you convert units, remember that lengths scale by the conversion factor, areas by its square and volumes by its cube:

• $1$ cm $=10$ mm, so $1$ cm$^2=100$ mm$^2$ and $1$ cm$^3=1000$ mm$^3$.
• $1$ m $=100$ cm, so $1$ m$^3=1\,000\,000$ cm$^3$.
• $1$ litre $=1000$ cm$^3$, and $1$ m$^3=1000$ litres. Make sure every length is in the same unit before you substitute into a formula.

A prism has the same cross-section all the way along its length. For any prism: $V=\text{area of cross-section}\times\text{length}.$ For a cylinder with radius $r$ and height $h$: $V=\pi r^2h,\qquad \text{curved surface area}=2\pi rh,\qquad \text{total surface area}=2\pi r^2+2\pi rh.$ For the surface area of a prism, find the area of each face and add. Example: a triangular prism with a right-angled triangle ($6$ cm, $8$ cm, $10$ cm) of length $15$ cm has $V=24\times15=360$ cm$^3$ and surface area $48+360=408$ cm$^2$.

A pyramid has a polygon base and triangular faces meeting at an apex. A cone has a circular base. Both have volume that is one third of the matching prism or cylinder: $V_{\text{pyramid}}=\frac13\times\text{base area}\times h,\qquad V_{\text{cone}}=\frac13\pi r^2h.$ Here $h$ is the perpendicular height. The slant height $l$ runs along a sloping face. For a cone, curved surface area $=\pi rl$. For a pyramid, each triangular face is $\frac12\times\text{base}\times\text{slant height}$. Example: base $10$ cm square, $h=12$ cm gives $V=\frac13(100)(12)=400$ cm$^3$.

A sphere of radius $r$ has $V=\frac43\pi r^3,\qquad \text{surface area}=4\pi r^2.$ A hemisphere is half a sphere. Its curved surface is $2\pi r^2$ and its total surface area, with the flat circle, is $3\pi r^2$. Example: $r=3$ cm gives $V=36\pi=113$ cm$^3$ and $A=36\pi=113$ cm$^2$ (the numbers match here, but the units differ). The formulae are given in the formula booklet, so focus on substituting carefully.

Many problems combine formulae. Plan the steps:

1. Pick the formula and write it down.
2. Check all lengths are in the same unit.
3. Substitute, keeping $\pi$ on your calculator until the end.
4. Round to 3 significant figures and give the correct units (cm$^3$ or cm$^2$). For liquids poured between containers, the volume stays the same. If the cup holds $100\pi$ cm$^3$ and is poured into a cylinder of radius $4$ cm, then $16\pi h=100\pi$ and $h=6.25$ cm. In real-life problems, round down for 'complete' items (servings, boxes) and round up for 'enough to buy'.