SURFACE AREAS AND VOLUMES

This chapter provides a detailed study of the surface areas and volumes of right circular cones and spheres (including hemispheres), building upon previous knowledge of cubes, cuboids, and cylinders.

1. Right Circular Cone

• Definition and Structure: A right circular cone is a solid generated by rotating a right-angled triangle about one of its perpendicular sides. It consists of a vertex, a circular base, a height (h), a radius (r), and a slant height (l).
• Slant Height Relationship: The relationship between the height, radius, and slant height is derived using the Pythagoras Theorem: l² = r² + h² or l = √(r² + h²).
• Curved Surface Area (CSA): The area of the lateral surface of a cone is given by the formula: πrl.
• Total Surface Area (TSA): This includes both the curved surface and the area of the circular base: πrl + πr², which simplifies to πr(l + r).
• Volume: The volume of a cone is exactly one-third the volume of a cylinder with the same base radius and height: (1/3)πr²h.

2. Sphere

• Definition: A sphere is a three-dimensional figure made of all points in space that lie at a constant distance (radius) from a fixed point (centre). It has only one curved face.
• Surface Area: The surface area of a sphere of radius r is four times the area of a circle with the same radius: 4πr².
• Volume: Based on water displacement principles, the volume of a sphere is given by: (4/3)πr³.

3. Hemisphere

• Definition: Slicing a solid sphere through the middle with a plane passing through its centre results in two equal parts called hemispheres.
• Curved Surface Area (CSA): Being half of the sphere's surface, the CSA is: 2πr².
• Total Surface Area (TSA): This includes the curved surface plus the flat circular base: 2πr² + πr² = 3πr².
• Volume: The volume of a hemisphere is half that of a sphere: (2/3)πr³.