Combined Solids: Solving Complex Problems

1. The Core Logic of Combining Solids

When we join two solid shapes, some surfaces get hidden. This is the most important concept for this lesson.

• For Volume: The logic is simple. Volume is about capacity. If you join two solids, the total volume is simply the sum of their individual volumes. If you carve a shape out of a larger solid, the remaining volume is the difference.

  • Volume_total = Volume_A + Volume_B
  • Volume_remaining = Volume_Outer - Volume_Inner

• For Surface Area: The logic is different. Surface area is about the exposed "skin" of the object. When you join two solids, the surfaces that touch are no longer exposed.

  • Incorrect: TSA_total ≠ TSA_A + TSA_B
  • Correct: TSA_total = Exposed Area of A + Exposed Area of B

This usually means adding the Curved/Lateral Surface Areas (CSA/LSA) of the component shapes and any exposed bases.

Let's look at a typical composite solid to make this clear.

If you were to paint the outside of this "Cone on a Cube" object, you would paint:

• The Curved Surface Area (CSA) of the cone.
• The five exposed faces of the cube (bottom and four sides).
• The top face of the cube MINUS the circular area covered by the cone's base.

You would NOT paint the base of the cone or the part of the cube it's sitting on. This is the central idea we will use today.

2. Calculating Surface Area of Combined Solids

Calculating surface area is a process of visualization. You must break the composite shape down and ask, "Which surfaces are on the outside?"

The following resource provides an excellent breakdown of this logic for various combinations.

Let's work through a classic example: A tent in the shape of a cylinder surmounted by a cone.

This image shows a composite solid formed by a cone on top of a cylinder, with their bases joined.

To find the total canvas required (i.e., the Total Surface Area), we need:

1. The Curved Surface Area (CSA) of the cylinder (the walls of the tent).
2. The Curved Surface Area (CSA) of the cone (the roof of the tent).
3. The Area of the base of the cylinder (the floor of the tent).

The circular top of the cylinder and the circular base of the cone are joined together, so they are not part of the external surface area.

Therefore, the formula is:

where $r$ is the common radius, $h$ is the height of the cylinder, and $l$ is the slant height of the cone.

3. Calculating Volume of Combined Solids

As we discussed, calculating volume is more straightforward. You just add or subtract the volumes of the component parts.

Let's see this in action with some worked examples. The following resource shows both an addition and a subtraction case.

Let's summarize the logic from those examples:

• Case 1: Addition (e.g., A pyramid on a cube)

  • The total volume is simply the volume of the cube plus the volume of the pyramid.
  • V_total = V_cube + V_pyramid = s³ + (1/3) * (Base Area) * h

• Case 2: Subtraction (e.g., A hemispherical scoop from a cylinder)

  • The remaining volume is the volume of the cylinder minus the volume of the hemisphere.
  • V_remaining = V_cylinder - V_hemisphere = (πr²h) - (2/3)πr³

4. Full Practice Problem

Let's tackle a problem from start to finish that requires both surface area and volume calculations.

Problem: A toy is in the shape of a cone of radius 7 cm mounted on a hemisphere of the same radius. The total height of the toy is 31 cm. Find the total surface area and the volume of the toy. (Use $\pi =22/7$)

Solution Steps:

1. Deconstruct the object and its dimensions:

   • The toy is a cone + hemisphere.
   • Radius $r=7$ cm for both parts.
   • Total height = 31 cm. The height of the hemispherical part is equal to its radius, so height of hemisphere = 7 cm.
   • Height of cone, $h_{cone}$ = Total Height - Height of hemisphere = 31 - 7 = 24 cm.

2. Calculate the Surface Area:

   • TSA = (CSA of cone) + (CSA of hemisphere).
   • We need the slant height ($l$) of the cone first: $l=\sqrt{r^2+h^2}=\sqrt{7^2+24^2}=\sqrt{49+576}=\sqrt{625}=25$ cm.
   • CSA of cone = $\pi rl=\frac{22}{7}\times 7\times 25=550$ cm².
   • CSA of hemisphere = $2\pi r^2=2\times \frac{22}{7}\times 7^2=2\times 22\times 7=308$ cm².
   • Total Surface Area = 550 + 308 = 858 cm².

3. Calculate the Volume:

   • Volume = (Volume of cone) + (Volume of hemisphere).
   • Volume of cone = $\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times \frac{22}{7}\times 7^2\times 24=\frac{1}{3}\times 22\times 7\times 24=22\times 7\times 8=1232$ cm³.
   • Volume of hemisphere = $\frac{2}{3}\pi r^3=\frac{2}{3}\times \frac{22}{7}\times 7^3=\frac{2}{3}\times 22\times 49=\frac{2156}{3}$ cm³.
   • Total Volume = $1232+\frac{2156}{3}=\frac{3696+2156}{3}=\frac{5852}{3}\approx 1950.67$ cm³.

Conclusion

Today we took our knowledge of basic 3D shapes and applied it to more complex, combined solids. The main challenge, and our primary focus, was understanding the different logic required for calculating surface area versus volume.

Key Takeaways:

• For Volume, think "Capacity": Simply add the volumes of the components, or subtract the volume of a part that has been carved out. It's an additive process.
• For Surface Area, think "Skin": Only add the areas of the exposed surfaces. This often involves using CSAs and ignoring the internal faces where solids are joined.
• Visualize First: Before writing any formulas, take a moment to visualize the composite solid and mentally list the surfaces you need to calculate.

Preview of the Next Lesson:
We have now concluded our study of 3D solids. In the next lesson, we will shift gears and enter the world of Coordinate Geometry. We will start with its most fundamental building blocks: learning how to apply the distance and section formulas to solve problems on the 2D coordinate plane. This will connect our understanding of geometry with algebra.