Perimeter, Area, and Volume

1. 2D Shapes

Core Area and Perimeter

Shape Perimeter Area
Rectangle $2(l + w)$ $l \times w$
Triangle - $\frac{1}{2} \times \text{base} \times \text{height}$
Parallelogram - $\text{base} \times \text{height}$
Trapezium - $\frac{1}{2}(a + b)h$

Circles

  • Circumference: $C = 2\pi r$ or $\pi d$.
  • Area: $A = \pi r^2$.
  • Arc Length: $L = \frac{\theta}{360} \times 2\pi r$.
  • Sector Area: $A = \frac{\theta}{360} \times \pi r^2$.
    • Extended: For major sectors, $\theta$ is the reflex angle.

Circle sector and arc diagram

2. 3D Shapes

Surface Area and Volume

Shape Volume Surface Area
Cuboid $l \times w \times h$ $2(lw + lh + wh)$
Prism $\text{Cross-section Area} \times \text{length}$ -
Cylinder $\pi r^2 h$ $2\pi r^2 + 2\pi rh$
Sphere $\frac{4}{3}\pi r^3$ $4\pi r^2$
Pyramid $\frac{1}{3} \times \text{Base Area} \times h$ -
Cone $\frac{1}{3}\pi r^2 h$ $\pi r^2 + \pi rl$ (where $l$ is slant height)

Frustums

A frustum is the part of a cone or pyramid that remains after the top is cut off by a plane parallel to the base.

  • Volume: $V_{\text{frustum}} = V_{\text{large}} - V_{\text{small}}$.
  • Surface Area: Sum of base areas and the lateral area (curved surface of the original minus the removed part).

Frustum diagram

3. Compound Shapes

  • Perimeter: Sum of all external boundaries.
  • Area: Split the composite 2D shape into simple rectangles, triangles, or circles. Add or subtract areas as necessary.
  • Volume: Split the composite 3D solid into simpler prisms, cylinders, or spheres. Sum the individual volumes.