This lesson focuses on finding the total area covering the outside of three key 3D shapes: the square-based pyramid, the cone, and the sphere. The main skill is identifying which formula to use and finding the correct measurements, often using Pythagoras' Theorem.

1. Square-Based Right Pyramid

A right pyramid has its apex directly above the center of its square base. All four triangular faces are identical isosceles triangles.

• 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.

  • Slant Height (l): The height of one of the triangular faces. This is not the same as 'h'!

• Formula for Total Surface Area (A): A = (Area of Square Base) + (Area of 4 Triangular Faces) A = a² + 4 (½ a * l) A = a² + 2al

• Key Skill: Finding the Slant Height (l) Most exam questions will give you 'h' and 'a', but you need 'l' for the formula. You find it using Pythagoras' Theorem on the internal right-angled triangle.

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

  • Example: If a pyramid has h=4cm and a=6cm, find 'l'.

    1. The base of the internal triangle is a/2 = 6/2 = 3cm.

    2. l² = 4² + 3² = 16 + 9 = 25

    3. l = √25 = 5 cm.

2. Right Circular Cone

• 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): The distance from the apex to any point on the edge of the circular base.

• Formulas:

  • Area of Curved Surface: A = πrl

  • Area of Circular Base: A = πr²

  • Total Surface Area: A = πrl + πr²

• Key Skill: Finding the Slant Height (l) Similar to the pyramid, you often need to find 'l' first using 'h' and 'r'.

  • The relationship is: l² = h² + r²

3. Sphere and Hemisphere

a) Sphere

A sphere has only one continuous curved surface.

• Formula for Surface Area: A = 4πr²

b) Hemisphere (Half-Sphere)

This is a common trick question area. A solid hemisphere has a curved part and a flat circular top.

• Area of Curved Surface: Half of a sphere's area -> ½ * 4πr² = 2πr²

• Area of Flat Circular Base: πr²

• Total Surface Area of a SOLID Hemisphere: A = (Curved Surface) + (Flat Base) A = 2πr² + πr² = 3πr²

Exam Tips & Common Mistakes

• Mistake 1: Confusing Heights! Do not mix up the perpendicular height (h) and the slant height (l). The formula for the surface area of pyramids and cones always uses the slant height (l). h is only used to find l via Pythagoras' Theorem.

• Mistake 2: Total vs. Curved Area. Read the question carefully. If it asks for the "area of the curved surface", don't include the base. If it says "total surface area", you must include the base.

• Mistake 3: Hemisphere Area. Forgetting to add the flat circular base (πr²) for a solid hemisphere is a very common error. The total surface area is 3πr², not 2πr².

• Exam Tip: For problems about tents or containers, they often don't have a base. The question might ask for the amount of "fabric needed for the roof", which means you only calculate the area of the triangular/curved surfaces.

• Value of Pi (π): Use 22/7 unless the question tells you otherwise (e.g., use 3.142). Using 22/7 often makes calculations easier, especially if the radius is a multiple of 7.