Split the compound shape into simple shapes you already have formulas for (rectangles, triangles, sectors, etc.), find each piece separately, then add the areas together for total area — but be careful with perimeter, which only uses the outer boundary, so any internal joining edges between the split pieces must be excluded.

Same idea in 3D: split the solid into standard solids (e.g. a cylinder topped with a hemisphere), calculate each volume separately and add them. For surface area, only include the faces that are actually exposed on the outside of the compound solid — any internal face where two solids join is hidden and must be left out.

Calculate the full shape's area/volume/surface area first using the standard formula, then multiply by the appropriate fraction ($\tfrac12$ for a half, $\tfrac14$ for a quarter, etc.). Take care with surface area here specifically: cutting a solid in half exposes a brand-new flat face that wasn't part of the original curved surface, so it must be added on separately.

A frustum is a cone (or pyramid) with its top sliced off parallel to the base. The standard method: calculate the volume of the complete, uncut cone, calculate the volume of the small cone that was removed from the top, then subtract. The same subtract-the-small-cone approach works for the curved surface area; the frustum's total surface area then also needs the two circular end-faces (top and bottom) added on.

The small cone and the complete (uncut) cone are similar solids, since the cut is parallel to the base — so their radii and heights are all in the same ratio. If you know the frustum's two radii (top and bottom) and its own height, set up a scale-factor equation comparing the small cone's radius to the full cone's radius to find how the heights must also scale, then solve for the full cone's height before subtracting.

A solid hemisphere has radius $6\text{ cm}$. Find its total surface area in terms of $\pi$.

1. Curved surface area of the hemisphere is half the sphere's surface area: $\tfrac12 \times 4\pi r^2 = \tfrac12 \times 4\pi(6)^2 = \tfrac12 \times 144\pi = 72\pi\text{ cm}^2$.
2. The flat circular face exposed by the cut has area $\pi r^2 = \pi(6)^2 = 36\pi\text{ cm}^2$.
3. Total surface area $= 72\pi + 36\pi = 108\pi\text{ cm}^2$.

A cone with base radius $8\text{ cm}$ and height $12\text{ cm}$ has its top sliced off parallel to the base, leaving a frustum with a top radius of $4\text{ cm}$. Find the volume of the frustum.

1. The small removed cone is similar to the full cone, with radius scale factor $\dfrac{4}{8} = \dfrac12$, so its height is also $\dfrac12 \times 12 = 6\text{ cm}$.
2. Volume of full cone $= \tfrac13 \pi (8)^2 (12) = 256\pi\text{ cm}^3$.
3. Volume of small removed cone $= \tfrac13 \pi (4)^2(6) = 32\pi\text{ cm}^3$.
4. Volume of frustum $= 256\pi - 32\pi = 224\pi \text{ cm}^3 \approx 704\text{ cm}^3$ (3 s.f.).

Label each piece on a sketch before calculating anything, decide whether each contributes positively (part of the shape) or needs subtracting (a hole or removed section), and total the areas or volumes only at the very end. Working in this order — labelling and planning first — avoids the common error of forgetting a piece or double-counting a shared edge partway through a longer calculation.