C11 Solid Figures
对应大纲的内容
| M5.7 | Know and use the formula for Pythagoras’ theorem: $a^{2} + b^{2} = c^{2}$
Use Pythagoras’ theorem in both 2 and 3 dimensions. |
| M5.11 | Know the terminology faces, surfaces, edges and vertices when applied to cubes, cuboids, prisms, cylinders, pyramids, cones, spheres and hemispheres. |
| M5.12 | Interpret plans and elevations of 3-dimensional shapes. |
| M5.14 | Know and apply formulae to calculate:
a. the volume of cuboids and other right prisms.
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| M5.15 | Know the formulae:
a. volume of a right circular cylinder = $\pi r^{2}h$
Formulae relating to spheres, pyramids and cones will be given if needed. Use formulae to calculate: b. surface area and volume of spheres, pyramids, cones and composite solids
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| M5.17 | Apply the concepts of congruence and similarity in simple figures, including the relationships between lengths, areas and volumes. |
| M5.18 | Know and use the trigonometric ratios:
$\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$ $\quad\cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$ $\quad\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}$ Apply these to find angles and lengths in right-angled triangles and, where possible, general triangles in 2- and 3-dimensional figures. |
学习目标
- Deconstruct 3D structures: Translate complex solids (like pyramids or tetrahedrons) into 2D cross-sections to apply Pythagoras’ theorem and trigonometry accurately.
- Master surface unrolling: Project 3D surfaces into 2D nets—such as flattening a cone into a sector—to calculate shortest-path surface trajectories.
- Synthesise composite volumes: Formulate algebraic relations for complex or nested solids (e.g., spheres inside cylinders) by applying standard volume and surface area formulae.
- Apply scaling laws: Use the foundational 3D scaling axioms: if the linear ratio is $k$, the area ratio is $k^{2}$ and the volume ratio is $k^{3}$.
- Decode projections: Reconstruct 3D geometry and block counts by interpreting 2D plans, such as top-down views and side elevations.
建议时间
| Task | No. of Questions | Suggested Time |
|---|---|---|
| Examples | 15 | 120 min |
| Quiz | 11 | 40 min |
| Practice A | 30 | 50 min |
| Practice B | 25 | 75 min |
| Total | 81 | approx. 4-5 hrs |
