U7 Area and Volume (I) Essential Formulas

This section covers the fundamental formulas for calculating the area of plane figures and the volume and surface area of common 3D solids. Mastery of these formulas is crucial for solving DSE problems.

1 Area of Plane Figures

Triangle

The area $A$ of a triangle can be found using base $b$ and height $h$, or using two sides $a$, $b$ and the included angle $\theta$.

$$ A = \frac{1}{2}bh = \frac{1}{2}ab\sin\theta $$

Circle

The area $A$ and circumference $C$ of a circle with radius $r$.

$$ A = \pi r^{2}, \quad C = 2\pi r $$

2 Volume and Surface Area of Solids

Prism and Cylinder

For any prism (including a cylinder), volume is base area $A$ times height $h$. The curved surface area of a right circular cylinder is $2\pi rh$.

$$ V = Ah $$

Prism

Cylinder

Pyramid and Cone

For any pyramid (including a cone), volume is one-third of base area $A$ times height $h$. The slant height $l$ is used for curved surface area.

$$ V = \frac{1}{3}Ah $$

Pyramid

Cone

Sphere

The volume $V$ and surface area $S$ of a sphere with radius $r$.

$$ V = \frac{4}{3}\pi r^{3}, \quad S = 4\pi r^{2} $$

3 Similar Figures and Solids

Linear, Area, and Volume Scale Factors

If two figures or solids are similar and their linear scale factor is $k$, then their area ratio is $k^{2}$ and their volume ratio is $k^{3}$.

$$ \frac{A_{1}}{A_{2}} = k^{2}, \quad \frac{V_{1}}{V_{2}} = k^{3} $$

For example, if a model car is $\frac{1}{10}$ the length of the real car ($k=\frac{1}{10}$), then its surface area is $\left(\frac{1}{10}\right)^{2} = \frac{1}{100}$ of the real car's, and its volume (and weight, if same material) is $\left(\frac{1}{10}\right)^{3} = \frac{1}{1000}$.

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