This lesson is about calculating the amount of space inside a 3D object. The key is remembering the correct formula and, most importantly, using the perpendicular height (h), not the slant height (l).

1. Square-Based Right Pyramid

• The Concept: The volume of a pyramid is exactly one-third the volume of a cuboid that has the same base and height.

• Parts of the Pyramid:

  • Base side (a): The length of one side of the square base.

  • Perpendicular Height (h): The height from the center of the base to the apex.

• Formula for Volume (V): V = ⅓ (Area of Base) Perpendicular Height V = ⅓ a² h

• Key Skill: Finding the Perpendicular Height (h) Sometimes, the exam will give you the slant height (l) and ask for the volume. You must first find 'h' using Pythagoras' Theorem.

  • The relationship is: l² = h² + (a/2)²

  • To find h: h = √[l² - (a/2)²]

2. Right Circular Cone

• The Concept: The volume of a cone is exactly one-third the volume of a cylinder that has the same base radius and height.

• Parts of the Cone:

  • Radius (r): The radius of the circular base.

  • Perpendicular Height (h): The height from the center of the base to the apex.

  • Slant Height (l): Used for surface area, not volume!

• Formula for Volume (V): V = ⅓ (Area of Base) Perpendicular Height V = ⅓ * πr²h

• Key Skill: Finding the Perpendicular Height (h) If you are given the slant height (l) and radius (r), you must find 'h' first.

  • The relationship is: l² = h² + r²

  • To find h: h = √[l² - r²]

3. Sphere and Hemisphere

a) Sphere

• The Concept: Archimedes discovered a famous relationship: the volume of a sphere is two-thirds the volume of its "circumscribing cylinder" (a cylinder that perfectly fits the sphere).

• Formula for Volume (V): V = (4/3)πr³

b) Hemisphere (Half-Sphere)

• Formula for Volume (V): A hemisphere is simply half of a sphere. V = ½ (Volume of Sphere) V = ½ (4/3)πr³ V = (2/3)πr³

Exam Tips & Common Mistakes

• Mistake 1: Using the Wrong Height! Volume formulas for pyramids and cones ALWAYS use the perpendicular height (h). The slant height (l) is for surface area. If the question gives you l, your first step must be to calculate h.

• Mistake 2: Forgetting the ⅓! This is the most common mistake for pyramids and cones. Remember, they are one-third of their "straight" counterparts (cuboid and cylinder).

• Mistake 3: Squaring instead of Cubing for Spheres. The formula for the surface area of a sphere is 4πr². The formula for volume is (4/3)πr³. Remember: area is in square units (cm²), so it uses r². Volume is in cubic units (cm³), so it uses r³.

• Exam Tip 1: "Melt and Recast" problems. If a question says a solid (e.g., a metal sphere) is melted and recast into another shape (e.g., several small cones), it means their VOLUMES are EQUAL.

  • How to solve:

    1. Calculate the volume of the original object.

    2. Calculate the volume of one of the new objects (with one variable, like 'h' or 'r', unknown).

    3. Set them equal to each other to solve for the unknown. Volume of Sphere = n * Volume of one Cone.

• Exam Tip 2: Diameter vs. Radius. Always check if the question gives you the diameter. If it does, halve it immediately to find the radius before you use any formula.