Introduction
This chapter deals with finding the surface areas and volumes of combinations of solids and the conversion of one solid into another. You will learn to calculate the surface area and volume when basic solids (cube, cuboid, cylinder, cone, sphere, hemisphere) are combined, and to find the resulting shape when one solid is melted or reshaped into another.
Surface Area of Combination of Solids
When two or more basic solids are combined, the surface area of the resulting figure is found by adding the surface areas of the individual parts, excluding the areas where they join. The total surface area is the sum of all exposed surfaces only. For example, when a cone is placed on top of a cylinder, the base of the cone and the top of the cylinder are not exposed.
Key Points
- •TSA of combination = Sum of visible surface areas only
- •Subtract areas where solids join (common surfaces)
- •A toy shaped as a cone on a hemisphere: TSA = CSA of cone + CSA of hemisphere
- •A capsule shape: TSA = CSA of cylinder + 2 x CSA of hemisphere
- •Always identify which surfaces are hidden (in contact)
Worked Example
A solid toy is in the form of a hemisphere surmounted by a right circular cone. Height of cone = 2 cm, radius = 1 cm. Find TSA. Slant height of cone l = sqrt(h^2 + r^2) = sqrt(4 + 1) = sqrt(5) cm CSA of cone = pi.r.l = pi(1)(sqrt(5)) = pi.sqrt(5) sq cm CSA of hemisphere = 2.pi.r^2 = 2.pi(1) = 2.pi sq cm TSA = pi.sqrt(5) + 2.pi = pi(sqrt(5) + 2) sq cm
Watch Out
Draw the combined solid and shade the hidden surfaces. This helps you identify which areas to exclude.
Volume of Combination of Solids
The volume of a combination of solids is simply the sum of the volumes of the individual parts. Unlike surface area, no subtraction is needed because volume is additive — the total space occupied is the sum of all component volumes.
Key Points
- •Volume of combination = Sum of individual volumes
- •No subtraction needed (unlike surface area)
- •Volume of cylinder = pi.r^2.h
- •Volume of cone = (1/3).pi.r^2.h
- •Volume of sphere = (4/3).pi.r^3, hemisphere = (2/3).pi.r^3
Worked Example
A gulab jamun is modelled as a cylinder of length 5 cm with two hemispheres of radius 1.4 cm attached at each end. Find volume. Length of cylindrical part = 5 - 2(1.4) = 2.2 cm Volume of cylinder = pi(1.4)^2(2.2) = pi(1.96)(2.2) = 4.312.pi cu cm Volume of 2 hemispheres = 2 x (2/3).pi(1.4)^3 = (4/3).pi(2.744) = 3.659.pi cu cm Total volume = 4.312.pi + 3.659.pi = 7.971.pi cu cm
Conversion of One Solid into Another
When a solid is melted and recast into another shape, the volume remains the same. This principle is used to find dimensions of the new shape. For example, if a sphere is melted and recast into small cylinders, the volume of the sphere equals the total volume of all cylinders.
Key Points
- •Volume of original solid = Volume of new solid(s)
- •If n identical solids are formed: n x volume of one = volume of original
- •Volume is conserved in melting/reshaping, surface area is NOT
- •Common problems: sphere to cylinder, cone to sphere, etc.
- •Number of objects = Volume of original / Volume of one object
Worked Example
A metallic sphere of radius 6 cm is melted into small cones of radius 2 cm and height 3 cm. Find the number of cones. Volume of sphere = (4/3).pi(6)^3 = 288.pi cu cm Volume of one cone = (1/3).pi(2)^2(3) = 4.pi cu cm Number of cones = 288.pi / 4.pi = 72
Basic Formulas for Standard Solids
It is essential to know the surface area and volume formulas for all standard solids: cube, cuboid, cylinder, cone, sphere, and hemisphere. These form the building blocks for all combination and conversion problems in this chapter.
Key Points
- •Cube: TSA = 6a^2, Volume = a^3
- •Cuboid: TSA = 2(lb + bh + hl), Volume = lbh
- •Cylinder: CSA = 2.pi.r.h, TSA = 2.pi.r(h+r), Volume = pi.r^2.h
- •Cone: CSA = pi.r.l, TSA = pi.r(l+r), Volume = (1/3).pi.r^2.h, l = sqrt(r^2+h^2)
- •Sphere: TSA = 4.pi.r^2, Volume = (4/3).pi.r^3
Quick Summary
- ✓TSA of combination = Sum of visible surfaces (exclude joined areas)
- ✓Volume of combination = Sum of individual volumes (no subtraction)
- ✓When reshaping: volume is conserved
- ✓Number of objects = Total volume / Volume of one object
- ✓Know all formulas for cube, cuboid, cylinder, cone, sphere, hemisphere
Key Formulas
Cylinder: CSA = 2.pi.r.h, TSA = 2.pi.r(h+r), V = pi.r^2.h
Cone: CSA = pi.r.l, TSA = pi.r(l+r), V = (1/3).pi.r^2.h, l = sqrt(r^2+h^2)
Sphere: TSA = 4.pi.r^2, V = (4/3).pi.r^3
Hemisphere: CSA = 2.pi.r^2, TSA = 3.pi.r^2, V = (2/3).pi.r^3
Cube: TSA = 6a^2, V = a^3
Cuboid: TSA = 2(lb+bh+hl), V = lbh
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