Area and Volume of Prisms
Area and Volume of Prisms
Area measures the amount of 2D surface. Volume measures the amount of 3D space enclosed. The formulae for triangles, parallelograms, trapezia and prisms (including cylinders) all derive from a small number of core principles.
Area Formulae
| Shape | Formula | Notes |
|---|---|---|
| Rectangle | \( A = lw \) | Base × height |
| Triangle | \( A = \frac{1}{2}bh \) | \( h \) = perpendicular height |
| Parallelogram | \( A = bh \) | \( h \) = perpendicular height (not slant) |
| Trapezium | \( A = \frac{1}{2}(a+b)h \) | \( a, b \) = parallel sides; \( h \) = perpendicular distance |
Volume of Prisms
Any prism (including a cylinder) has volume:
\[ V = A_{\text{cross-section}} \times l \]where \( l \) is the length (depth) of the prism and \( A_{\text{cross-section}} \) is the area of the uniform cross-section.
- Cuboid: \( V = lwh \) (cross-section is a rectangle)
- Triangular prism: \( V = \frac{1}{2}bhl \) (cross-section is a triangle)
- Cylinder: \( V = \pi r^2 h \) (cross-section is a circle)
- Trapezoidal prism: \( V = \frac{1}{2}(a+b)h \times l \)
Composite Shapes
For composite shapes, split into simpler components, calculate each area (or volume) separately, then add (or subtract for holes).
Worked Examples
Find the area of a trapezium with parallel sides 8 cm and 14 cm and perpendicular height 6 cm.
\[ A = \frac{1}{2}(8+14)\times 6 = \frac{1}{2}\times22\times6 = 66 \text{ cm}^2 \]A parallelogram has base 9 cm and perpendicular height 5 cm. Find its area.
\[ A = 9 \times 5 = 45 \text{ cm}^2 \]A triangular prism has a right-angled triangular cross-section with legs 6 cm and 8 cm. The prism is 15 cm long. Find its volume.
\[ A_{\triangle} = \frac{1}{2}\times6\times8 = 24 \text{ cm}^2 \qquad V = 24\times15 = 360 \text{ cm}^3 \]A cylinder has radius 4 cm and height 10 cm. Find its volume (leave in terms of \( \pi \)).
\[ V = \pi\times4^2\times10 = 160\pi \text{ cm}^3 \]An L-shaped cross-section prism: the L consists of a 10×8 rectangle with a 4×5 rectangle removed from one corner. The prism is 6 cm deep. Find its volume.
\[ A_{\text{L}} = 10\times8 - 4\times5 = 80-20 = 60 \text{ cm}^2 \qquad V = 60\times6 = 360 \text{ cm}^3 \]Key Takeaways
- Trapezium: \( \frac{1}{2}(a+b)h \). The \( h \) is always the perpendicular distance between the parallel sides.
- All prism volumes: area of cross-section × length. Identify the cross-section first.
- Cylinder: special case — cross-section is a circle, so \( V=\pi r^2 h \).
- Perpendicular height, not slant height, in all area and volume formulae.
- Composite volumes: split, calculate parts, add (or subtract for holes/cutouts).