E5.5 Compound shapes and parts of shapes

First mark missing lengths using totals, equal sides, radii or diameters. Choose a decomposition whose pieces do not overlap. Write every component area before adding or subtracting; for a shaded region, identify clearly which area is removed. For perimeter, start at one point and travel once around the finished shape, recording each boundary segment in order. Use arc length—not circle area—for curved edges, and keep units consistent.

A $10$ cm by $6$ cm rectangle has a $4$ cm by $2$ cm corner removed. The remaining area is $10\times6-4\times2=52\text{ cm}^2$. Tracing its six outside edges gives $10+4+4+2+6+6=32$ cm. The two cut edges must be included because they become exposed, but the removed corner is not part of the area.

Draw up a component ledger before calculating. Keep one unit throughout, use each full-solid formula, and preserve $\pi$ until the end. A join does not change total volume, but it hides matching surfaces. For a frustum, extend it mentally to the original cone or pyramid; use similarity to find the removed top's dimensions, then subtract its volume. Lengths scale by $k$, areas by $k^2$ and volumes by $k^3$.

A cone of height $12$ cm and radius $6$ cm is cut parallel to its base so the removed small cone has height $4$ cm. Similarity gives its radius as $6\times4/12=2$ cm. Frustum volume $=\frac13\pi(6)^2(12)-\frac13\pi(2)^2(4)=\frac{416\pi}{3}\text{ cm}^3$. For surface area, count the outer curved frustum and both exposed circular ends, not the removed cone's base as an internal join.