3D shapes and volumes

Mark scheme

1. Answer. $60$ cm$^{3}$

   • $3 \times 4 \times 5$
   • $60$ cm$^{3}$

2. Answer. $96$ cm$^{2}$

   • One face has area $16$.
   • $6 \times 16 = 96$ cm$^{2}$

3. Answer. $90\pi$ cm$^{3}$

   • $V = \pi r^{2} h$
   • $\pi \times 9 \times 10 = 90\pi$ cm$^{3}$

4. Answer. $96$ cm$^{3}$

   • Volume $= \text{area of cross-section} \times \text{length}$
   • $96$ cm$^{3}$

5. Answer. $7$ cm

   • $\sqrt{2^{2} + 3^{2} + 6^{2}} = \sqrt{49}$
   • $7$ cm

6. Answer. Volume $12\pi$ cm$^{3}$, surface area $24\pi$ cm$^{2}$

   • Volume $= \dfrac{1}{3}\pi r^{2} h = 12\pi$ cm$^{3}$.
   • Slant height $= \sqrt{3^{2} + 4^{2}} = 5$ cm.
   • Curved area $= \pi \times 3 \times 5 = 15\pi$, plus base $9\pi$, gives $24\pi$ cm$^{2}$.

7. Answer. Volume $120$ cm$^{3}$, surface area $158$ cm$^{2}$, diagonal $7\sqrt{2}$ cm

   • Volume $= 8 \times 5 \times 3 = 120$ cm$^{3}$.
   • Surface area $= 2(40 + 24 + 15) = 158$ cm$^{2}$.
   • Diagonal $= \sqrt{64 + 25 + 9} = \sqrt{98} = 7\sqrt{2}$ cm.

8. Answer. Cylinder $300\pi$ cm$^{3}$, sphere $\dfrac{500\pi}{3}$ cm$^{3}$, difference $\dfrac{400\pi}{3}$ cm$^{3}$

   • Cylinder $= \pi \times 25 \times 12 = 300\pi$ cm$^{3}$.
   • Sphere $= \dfrac{4}{3}\pi \times 125 = \dfrac{500\pi}{3}$ cm$^{3}$.
   • Difference $= 300\pi - \dfrac{500\pi}{3} = \dfrac{400\pi}{3}$ cm$^{3}$.