| Solid | Curved/Lateral Area (CSA) | Total Surface Area (TSA) | Volume ($V$) |
|---|---|---|---|
| Cuboid | $2h(l + b)$ | $2(lb + bh + hl)$ | $l \times b \times h$ |
| Cube | $4a^2$ | $6a^2$ | $a^3$ |
| Cylinder | $2\pi rh$ | $2\pi r(r + h)$ | $\pi r^2 h$ |
| Cone | $\pi r l \quad (l = \sqrt{r^2 + h^2})$ | $\pi r(l + r)$ | $\frac{1}{3}\pi r^2 h$ |
| Sphere | $4\pi r^2$ | $4\pi r^2$ | $\frac{4}{3}\pi r^3$ |
| Hemisphere | $2\pi r^2$ | $3\pi r^2$ | $\frac{2}{3}\pi r^3$ |
A toy is in the form of a cone of radius $3.5\text{ cm}$ mounted on a hemisphere of same radius. The total height of the toy is $15.5\text{ cm}$. Find the total surface area of the toy.
Solution:
Given:
Radius of hemisphere and cone ($r$) = $3.5\text{ cm} = \frac{7}{2}\text{ cm}$.
Height of hemispherical part = $r = 3.5\text{ cm}$.
Height of conical part ($h$) = $15.5 - 3.5 = 12\text{ cm}$.
Slant height of cone ($l$):
$$l = \sqrt{r^2 + h^2}$$
$$l = \sqrt{(3.5)^2 + 12^2}$$
$$l = \sqrt{12.25 + 144} = \sqrt{156.25} = 12.5\text{ cm}$$
Total Surface Area of Toy = $\text{CSA of Cone} + \text{CSA of Hemisphere}$
$$\text{TSA} = \pi r l + 2\pi r^2 = \pi r(l + 2r)$$
$$\text{TSA} = \frac{22}{7} \times \frac{7}{2} \times \left(12.5 + 2(3.5)\right)$$
$$\text{TSA} = 11 \times (12.5 + 7)$$
$$\text{TSA} = 11 \times 19.5 = 214.5\text{ cm}^2$$
The total surface area of the toy is $214.5\text{ cm}^2$.
A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to $1\text{ cm}$ and the height of the cone is equal to its radius. Find the volume of the solid in terms of $\pi$.
Solution:
Given:
Radius ($r$) = $1\text{ cm}$
Height of cone ($h$) = $1\text{ cm}$
Volume of the solid = $\text{Volume of Cone} + \text{Volume of Hemisphere}$
$$V = \frac{1}{3}\pi r^2 h + \frac{2}{3}\pi r^3$$
$$V = \frac{1}{3}\pi (1)^2(1) + \frac{2}{3}\pi (1)^3$$
$$V = \frac{\pi}{3} + \frac{2\pi}{3}$$
$$V = \frac{3\pi}{3} = \pi\text{ cm}^3$$
The volume of the solid is $\pi\text{ cm}^3$.