U20 Area and Volume (II) Essential Formulas

This section covers advanced concepts in mensuration, focusing on the surface area and volume of spheres, cones, pyramids, and their frustums. Understanding the relationships between similar solids is crucial.

1 Sphere

Surface Area and Volume

For a sphere with radius $r$, its surface area $A$ and volume $V$ are given by the following formulas.

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

2 Right Circular Cone

Curved Surface Area and Volume

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

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

3 Frustum of a Cone

Volume and Curved Surface Area

A frustum is formed by cutting a cone with a plane parallel to its base. Let $R$ and $r$ be the radii of the lower and upper bases, $h$ be the height, and $\ell$ be the slant height.

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

4 Similar Solids

Ratio of Lengths, Areas, and Volumes

If two solids are similar and the ratio of their corresponding lengths is $k$, then the ratio of their areas is $k^2$, and the ratio of their volumes is $k^3$.

$$ \frac{A_1}{A_2} = k^2 \quad \text{and} \quad \frac{V_1}{V_2} = k^3 $$

This principle applies to all similar three-dimensional shapes, such as spheres, cubes, and similar cones.

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