Solid Geometry

1Summary 重點摘要

Prism, cylinder, pyramid, cone, sphere volume and surface area.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Volume of a prismA prism has two congruent, parallel polygonal bases and parallelogram side faces; its volume is V = base area × height. The height here means the perpendicular distance between the two bases, not the length of a slanted edge, which matters especially when the prism is oblique. Triangular prisms, cuboids and cubes are all prisms; a cuboid's volume of length × width × height is simply base area times height. When finding the base area, choose the area formula that matches the shape of the base.
2. Step 2: Volume of a cylinderA cylinder can be viewed as a prism with circular bases, so its volume is again base area times height: V = πr²h, where r is the base radius and h is the perpendicular height. Compute the base area πr² first and multiply by the height afterwards; reversing the order invites missing the square. Distinguish radius from diameter: if the diameter is given, halve it first. An oblique cylinder also uses the perpendicular height, for the same reason as an oblique prism, and the exam sometimes asks about this, so read carefully whether a radius or a diameter is given.
3. Step 3: Volume of a pyramidA pyramid has a single polygonal base with triangular faces meeting at one apex; its volume is V = (13) × base area × height. The height is the perpendicular distance from the apex to the base, not the slanted edge of a lateral face; in solid diagrams this height often has to be added as an auxiliary line and found with Pythagoras' theorem or trigonometry. The factor 13 arises because a pyramid holds exactly one third of the volume of the prism with the same base and height, which a water-or-sand demonstration shows. Forgetting the 13 is the most common error in this topic, so always write the formula first.
4. Step 4: Volume of a coneA cone is a pointed solid with a circular base, and its volume is V = (13)πr²h, exactly one third of the cylinder with the same base and height, mirroring the pyramid-prism relationship. A clear order of work is: write the base area πr², multiply by the height, and then multiply by 13. The exam often pairs this with the surface area or the slant height l, and l, r and h satisfy l² = r² + h², which comes from the axial cross-section being an isosceles triangle that can be split into two right triangles.
5. Step 5: Volume of a sphereA sphere is perfectly symmetric: every point on its surface is the radius r away from the centre, and its volume is V = (43)πr³. The radius is cubed in the formula, so doubling the radius multiplies the volume by eight, a scaling fact often tested in multiple-choice questions. Compute r³ first, then multiply by 43 and by π. A sphere has no apex and no flat side faces, so prism or pyramid formulas do not apply; if the diameter is given, halve it to get the radius before substituting, a basic but frequent slip.
6. Step 6: Handling surface areaSurface area adds up every face of the solid: a prism is two bases plus the parallelogram sides; a cylinder is two circular bases (2πr²) plus the curved side (2πr × h, the base circumference times the height); a cone is the circular base plus the sector-shaped curved surface (πrl, with l the slant height); a sphere is 4πr². The strategy is to take the solid apart: picture its net, list every piece, compute each one and add them up. Typical errors are missing a hidden base, or giving the curved surface area when the total surface area is asked, so read carefully which one the question wants.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Prism volume = base area × height
• Pyramid volume = 13 × base area × height
• Sphere volume = 43 πr³
• Cylinder side = 2πr × height
• Surface area count all faces