5. Mensuration

To convert $0.17\text{ m}^2$ to $\text{cm}^2$, use $1\text{ m}^2=100^2\text{ cm}^2$: $0.17\times10\,000=1700\text{ cm}^2$. To convert $8500\text{ cm}^3$ to litres, use $1000\text{ cm}^3=1$ litre: $8500\div1000=8.5$ litres.

A trapezium has parallel sides $7$ cm and $11$ cm and perpendicular height $5$ cm. Its area is $\frac12(7+11)\times5=45\text{ cm}^2$. If a triangle has area $27\text{ cm}^2$ and base $6$ cm, then $\frac12\times6\times h=27$, so $h=9$ cm.

A circle has diameter $12$ cm. Then $r=6$ cm, so $C=12\pi$ cm and $A=36\pi\text{ cm}^2$. If instead its area is $150\text{ cm}^2$, then $r=\sqrt{150/\pi}=6.91\ldots$ cm. Use the requested accuracy; leave the answer in terms of $\pi$ when asked.

Use the angle at the centre. For a minor sector with angle $\theta$, use $\theta/360$; the matching major sector uses $(360-\theta)/360$. Choose circumference for an arc or circle area for a sector, then multiply. If total sector perimeter is required, add the two radii to the arc length; those straight edges are not part of the arc.

For $r=9$ cm and minor angle $140^\circ$, the minor arc length is $\frac{140}{360}(2\pi\times9)=7\pi$ cm. The minor sector area is $\frac{140}{360}(\pi\times9^2)=\frac{63\pi}{2}\text{ cm}^2$. The major angle is $220^\circ$, so the major sector area is $\frac{220}{360}(81\pi)=\frac{99\pi}{2}\text{ cm}^2$.

A closed cylinder has radius $3$ cm and height $8$ cm. Its total surface area is $2\pi(3)^2+2\pi(3)(8)=66\pi\text{ cm}^2$, while its volume is $\pi(3)^2(8)=72\pi\text{ cm}^3$. Leaving $\pi$ exact keeps both values accurate until a decimal is requested.

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.