Areas and volumes

Description

Area measures the amount of space inside a 2D shape (in square units). Volume measures the space inside a 3D shape (in cubic units).

Key area formulae:

• Rectangle: $A = lw$
• Triangle: $A = \frac{1}{2}bh$
• Parallelogram: $A = bh$
• Trapezium: $A = \frac{1}{2}(a+b)h$
• Circle: $A = \pi r^2$

Key volume formulae:

• Cuboid: $V = lwh$
• Prism: $V = \text{cross-sectional area} \times \text{length}$
• Cylinder: $V = \pi r^2 h$
• Cone: $V = \frac{1}{3}\pi r^2 h$
• Sphere: $V = \frac{4}{3}\pi r^3$
• Pyramid: $V = \frac{1}{3} \times \text{base area} \times h$

Surface area: sum of the areas of all faces.

Example: Volume of a cylinder with $r = 3$ and $h = 10$: $$V = \pi \times 9 \times 10 = 90\pi \approx 282.7\text{ cm}^3$$

Common error: using diameter instead of radius in circle or cylinder formulae; forgetting the $\frac{1}{3}$ in cone and pyramid volumes.

Links

Substitution

Use $\pi$ key throughout. Enter the full formula in one calculation with brackets to avoid errors.