Key Points

For a cuboid with length $l$, breadth $b$, and height $h$: Lateral Surface Area (LSA) is $2(l+b)h$, Total Surface Area (TSA) is $2(lb + bh + hl)$, and Volume is $V = l \times b \times h$.

For a cube with edge length $a$: Lateral Surface Area (LSA) is $4a^2$, Total Surface Area (TSA) is $6a^2$, and Volume is $V = a^3$.

For a right circular cylinder with radius $r$ and height $h$: Curved Surface Area (CSA) is $2\pi rh$, Total Surface Area (TSA) is $2\pi r(r+h)$, and Volume is $V = \pi r^2 h$.

For a right circular cone with radius $r$, height $h$, and slant height $l$: Curved Surface Area (CSA) is $\pi rl$, Total Surface Area (TSA) is $\pi r(r+l)$, and Volume is $V = \frac{1}{3}\pi r^2 h$.

The slant height $l$ of a cone is found using the Pythagorean theorem with its radius $r$ and height $h$. The formula is $l = \sqrt{r^2 + h^2}$.

For a sphere with radius $r$: Surface Area is $4\pi r^2$ and Volume is $V = \frac{4}{3}\pi r^3$. A sphere has only one continuous curved surface.

For a hemisphere with radius $r$: Curved Surface Area (CSA) is $2\pi r^2$, Total Surface Area (TSA) is $3\pi r^2$ (CSA + base area), and Volume is $V = \frac{2}{3}\pi r^3$.

To find the surface area of a combined solid, add the areas of all visible surfaces. Do not include the areas of the faces where the solids are joined together, as they are no longer exposed.

To find the volume of a combined solid, simply add the volumes of the individual component solids. If a solid is hollowed out, subtract the volume of the removed portion.

For a toy made of a cone mounted on a hemisphere of the same radius $r$, the total surface area is the sum of their curved surface areas. TSA = (CSA of cone) + (CSA of hemisphere) = $\pi rl + 2\pi r^2$.

A capsule is a cylinder with two hemispheres at its ends. Its total surface area is the CSA of the cylinder plus the CSA of the two hemispheres. TSA = $2\pi rh + 2(2\pi r^2) = 2\pi r(h + 2r)$.

The surface area is the TSA of the cube, minus the base area of the hemisphere (which is covered), plus the CSA of the hemisphere. TSA = $6(\text{edge})^2 - \pi r^2 + 2\pi r^2 = 6(\text{edge})^2 + \pi r^2$.

When a shape is hollowed out (e.g., a conical cavity from a cylinder), the surface area increases. The new total area = (CSA of original solid) + (Area of base) + (CSA of the hollowed-out shape).

The volume of a toy composed of a cone mounted on a hemisphere is the sum of their individual volumes. Total Volume = (Volume of cone) + (Volume of hemisphere) = $\frac{1}{3}\pi r^2 h + \frac{2}{3}\pi r^3$.

If a shape is hollowed out from a solid (e.g., conical depressions in a cuboid), the volume of the remaining solid is the volume of the original solid minus the volume of the removed parts.