U27 Area and Volume (III) Essential Formulas

This topic extends the concepts of area and volume to more complex 3D solids, focusing on spheres, cones, pyramids, and their frustums. Mastery of these formulas is crucial for solving DSE problems involving composite solids and rate of change.

1 Sphere and Hemisphere

Sphere

For a sphere with radius $r$, its surface area (or curved surface area) and volume are given by:

$$ A = 4\pi r^2 $$
$$ V = \frac{4}{3}\pi r^3 $$
$r$

Hemisphere

A hemisphere is half of a sphere. Its total surface area includes the curved surface and the circular base.

Total surface area = Curved surface area + Area of base.

$$ A_{\text{total}} = 2\pi r^2 + \pi r^2 = 3\pi r^2 $$
$$ V = \frac{1}{2} \times \frac{4}{3}\pi r^3 = \frac{2}{3}\pi r^3 $$

2 Right Circular Cone

Formulas

For a right circular cone with base radius $r$, height $h$, and slant height $\ell$, we have the relation $\ell = \sqrt{r^2 + h^2}$.

$$ \text{Curved Surface Area} = \pi r \ell $$
$$ \text{Total Surface Area} = \pi r \ell + \pi r^2 $$
$$ \text{Volume} = \frac{1}{3} \pi r^2 h $$
$h$$\ell$$r$

3 Pyramid and Frustum of a Cone

Right Pyramid

For any right pyramid (with a regular polygonal base), the volume is one-third of the product of the base area and the height.

$$ V = \frac{1}{3} \times \text{Base Area} \times h $$

Frustum of a Cone

A frustum is the part of a cone (or pyramid) left after cutting off the top by a plane parallel to the base. Let $R$ and $r$ be the radii of the lower and upper bases, $h$ be the height, and $\ell$ be the slant height where $\ell = \sqrt{h^2 + (R - r)^2}$.

$$ \text{Curved Surface Area} = \pi (R + r) \ell $$
$$ \text{Volume} = \frac{1}{3} \pi h (R^2 + Rr + r^2) $$

4 Similar Solids

Area and Volume Ratios

If two solids are similar and the ratio of their corresponding lengths is $k : 1$, then:

$$ \frac{\text{Area}_1}{\text{Area}_2} = k^2 $$
$$ \frac{\text{Volume}_1}{\text{Volume}_2} = k^3 $$

This principle is frequently applied to problems involving scaling models, rate of change of dimensions, and comparing parts of similar solids.

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