Summary: Surface Area And Volume Of Sphere

Table of Contents
1. Introduction
2. Surface Area - key ideas
3. Surface area formulas
4. Volume - key ideas
5. Volume formulas
6. Practical notes for combined solids
View more

Introduction

• Combined solids are shapes made by joining basic 3D figures (cube, cuboid, cylinder, cone, sphere, hemisphere).
• We find their surface area and volume by adding appropriate parts of the constituent shapes.

Surface Area - key ideas

• Total surface area (TSA) = sum of all outer surfaces including bases and curved parts.
• Curved surface area (CSA) or lateral surface area = area of only the curved part, excluding bases.
• For a combined solid, TSA or CSA is obtained by adding the required surface areas of each part, taking care to exclude any internal faces where parts join.

Surface area formulas

• Cube: TSA = 6a², where a = side.
• Cuboid: TSA = 2(lb + bh + lh), where l = length, b = breadth, h = height.
• Right circular cylinder: CSA = 2πrh, TSA = 2πr(h + r), where r = radius, h = height.
• Hollow right circular cylinder: CSA uses outer and inner circumferences appropriately (use difference of radii in curved area when needed).
• Right circular cone: CSA = πrl, TSA = πr(r + l), where l = slant height and l = √(h² + r²).
• Sphere: CSA = TSA = 4πr², where r = radius.
• Hemisphere: CSA (curved part) = 2πr²; TSA (including flat face) = 3πr².

Volume - key ideas

• The volume of a combined solid = sum of volumes of its constituent parts (add or subtract where parts are removed or hollow).

Volume formulas

• Cube: V = a³.
• Cuboid: V = lbh.
• Cylinder: V = πr²h.
• Cone: V = (1/3)πr²h; l = √(h² + r²).
• Sphere: V = (4/3)πr³.
• Hemisphere: V = (2/3)πr³.

Practical notes for combined solids

• Decide which surface areas or volumes of parts are needed (include outer faces, exclude internal joining faces).
• Use matching radii/heights when parts share dimensions; add volumes directly for joined parts.
• For hollow parts, subtract inner volume or use difference in curved areas as appropriate.