What is a combination of solids? A combination of solids is a complex three-dimensional object formed by joining two or more basic three-dimensional shapes together. In the real world, very few things are pure shapes like an isolated cylinder or a perfect sphere. For instance, an ice cream cone is a combination of a cone topped by a hemisphere. A medical capsule is a cylinder with hemispheres stuck to both ends. A circus tent often looks like a cylinder capped by a cone.
When analyzing these real-world objects, we need to find their total surface area and total volume by looking at the individual shapes that make them up.
Total Surface Area (TSA) of Combined Solids When two shapes are joined together tightly, some of their individual faces get trapped inside and disappear from view. Therefore, the total surface area of a combined solid is the sum of only the visible outer surfaces.
- Rule: Look only at what you can touch or paint from the outside.
- Example: For a toy made by joining a cone to the flat base of a hemisphere, the joining surfaces are hidden inside. The total surface area is equal to the Curved Surface Area (CSA) of the cone plus the Curved Surface Area (CSA) of the hemisphere.
Total Volume of Combined Solids Volume measures the space trapped inside an object. Space does not disappear when objects are joined together.
- Rule: The total volume is simply the sum of the volumes of all the individual shapes.
- Example: The volume of our ice cream cone toy is the volume of the cone plus the volume of the hemisphere.
| Shape Component | Curved Surface Area (CSA) | Total Surface Area (TSA) | Volume |
|---|---|---|---|
| Cube (side $a$) | $4a^2$ | $6a^2$ | $a^3$ |
| Cuboid ($l, w, h$) | $2h(l + w)$ | $2(lw + wh + hl)$ | $l \cdot w \cdot h$ |
| Cylinder ($r, h$) | $2\pi rh$ | $2\pi r(r + h)$ | $\pi r^2h$ |
| Cone ($r, h, slant\ l$) | $\pi rl$ | $\pi r(r + l)$ | $(1/3)\pi r^2h$ |
| Sphere ($r$) | $4\pi r^2$ | $4\pi r^2$ | $(4/3)\pi r^3$ |
| Hemisphere ($r$) | $2\pi r^2$ | $3\pi r^2$ | $(2/3)\pi r^3$ |
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