Chapter 12: Surface Areas and Volumes

So far we have measured flat shapes , circles, sectors, triangles, quadrilaterals. This chapter steps into three dimensions. Each solid is built up from flat pieces , a cylinder is two circles plus a rectangle wrapped around them, a cone is a circle plus a sector unrolled, a sphere is harder to unroll but obeys neat formulas of its own. The two main quantities we compute are surface area (the total flat area you would need to wrap the solid, measured in cm$^2$) and volume (the space the solid fills, measured in cm$^3$).

The chapter's signature problem is the combined solid: a cone on top of a cylinder, a hemisphere on the base of a cylinder, a frustum joining two circles of different sizes. To find the surface area or volume of a combination, you add the external surface areas and the full volumes of the pieces, taking care not to double-count surfaces that are hidden inside.

The applications are everywhere: an ice-cream cone with a scoop on top (cone + hemisphere), a water tanker (cylinder with hemispherical ends), a bucket (frustum), a tent (cone on cylinder), a child's toy. Board exam questions usually describe the solid in words, ask you to identify the parts, and then compute volume or surface area.

A key skill is converting volumes between shapes. If a metallic cone is melted and recast into a sphere, equating volumes gives the new radius. Hundreds of board problems use this idea.

What's inside

• Cuboid and cube , formulas review.
• Cylinder , curved + total surface area, volume.
• Cone , slant height, lateral surface area, volume.
• Sphere and hemisphere , surface area and volume.
• Frustum of a cone , surface area, volume.
• Combined solids , strategy and worked examples.

Key results / Formula card

For a cuboid with sides $\ell, b, h$: $V = \ell b h$, surface area $= 2(\ell b + bh + h\ell)$.

For a cube with side $a$: $V = a^3$, surface area $= 6 a^2$.

For a cylinder with radius $r$ and height $h$:

• Curved surface area: $2\pi r h$.
• Total surface area: $2\pi r h + 2\pi r^2 = 2\pi r(h + r)$.
• Volume: $\pi r^2 h$.

For a cone with radius $r$, height $h$, slant height $\ell = \sqrt{r^2 + h^2}$:

• Curved surface area: $\pi r \ell$.
• Total surface area: $\pi r \ell + \pi r^2 = \pi r(\ell + r)$.
• Volume: $(1/3) \pi r^2 h$.

For a sphere of radius $r$:

• Surface area: $4\pi r^2$.
• Volume: $(4/3) \pi r^3$.

For a hemisphere of radius $r$:

• Curved surface area: $2\pi r^2$.
• Total surface area: $3\pi r^2$.
• Volume: $(2/3) \pi r^3$.

For a frustum of a cone with radii $r_1, r_2$ and height $h$, slant $\ell = \sqrt{h^2 + (r_1 - r_2)^2}$:

• Curved surface area: $\pi(r_1 + r_2) \ell$.
• Total surface area: $\pi(r_1 + r_2)\ell + \pi r_1^2 + \pi r_2^2$.
• Volume: $(1/3) \pi h (r_1^2 + r_2^2 + r_1 r_2)$.