Learning Objectives
- Find surface areas and volumes of combinations of solids
- Convert one solid shape to another and find resulting dimensions
- Find the frustum of a cone: surface area and volume
Key Concepts
Surface Areas and Volumes of Basic Solids
- Cuboid: TSA = 2(lb + bh + hl), Volume = l × b × h
- Cube: TSA = 6a², Volume = a³
- Cylinder: CSA = 2πrh, TSA = 2πr(r + h), Volume = πr²h
- Cone: CSA = πrl, TSA = πr(r + l), Volume = ⅓πr²h, Slant height l = √(r² + h²)
- Sphere: TSA = 4πr², Volume = 4/3 πr³
- Hemisphere: CSA = 2πr², TSA = 3πr², Volume = 2/3 πr³
Combination of Solids
When two or more basic solids are joined, the total surface area is the sum of the visible surface areas of each solid (excluding the parts where they are joined).
The total volume is the sum of the individual volumes.
Conversion of Solids
When a solid is melted and recast into another shape, the volume remains constant. Use this to find unknown dimensions of the new solid.
Example: A cone melted and recast into a sphere → ⅓πr₁²h = 4/3 πr₂³
Frustum of a Cone
A frustum is formed when a cone is cut by a plane parallel to its base. Let R = radius of base, r = radius of top, h = height, l = slant height.
- Slant height: l = √[h² + (R - r)²]
- CSA = π(R + r)l
- TSA = π(R + r)l + πR² + πr²
- Volume = ⅓πh(R² + r² + Rr)
Summary
This chapter extends surface area and volume concepts to combinations of solids and frustums. When solids are combined, surface areas require careful identification of exposed surfaces. Volume is conserved when solids are reshaped. The frustum of a cone has its own set of formulas derived from the cone formulas.
Important Terms
- Frustum
- The portion of a cone between its base and a plane cutting it parallel to the base
- CSA (Curved Surface Area)
- The area of the curved portion of a solid, excluding flat bases
- TSA (Total Surface Area)
- The sum of all surface areas including the curved surface and flat bases
- Slant Height
- The distance along the surface of a cone or frustum from base edge to apex or top edge
Quick Revision
- Volume of sphere = 4/3 πr³; Volume of cone = 1/3 πr²h; Volume of cylinder = πr²h
- When reshaping, volume stays the same
- Frustum volume = ⅓πh(R² + r² + Rr)
- Frustum slant height = √[h² + (R-r)²]
- For combined solids: add volumes, but for surface area, subtract the joined surfaces