Circles and Composite Shapes

Circles and Composite Shapes

The circle formulae are among the most frequently used in GCSE geometry. Composite shapes combine circles, semicircles or quarter-circles with polygons — requiring the area and perimeter to be built from parts.

Core Circle Formulae

\[ C = 2\pi r = \pi d \qquad A = \pi r^2 \]

Always identify whether you are given the radius or the diameter before substituting.

Composite Shapes — Strategy

  1. Identify the component shapes (rectangle, triangle, semicircle, quarter-circle, etc.).
  2. Calculate the area (or perimeter contribution) of each component.
  3. Add for the total area. For perimeter, only count edges that form the outer boundary — internal joins are not part of the perimeter.
h d = 2r r (join — not on perimeter) perimeter

Worked Example

The composite shape above has a rectangle of width \( 2r \) and height \( h \), topped by a semicircle of radius \( r \).

\[ \text{Area} = 2rh + \frac{1}{2}\pi r^2 \] \[ \text{Perimeter} = 2h + 2r + \pi r \quad \text{(two sides of rectangle + bottom + semicircle arc; top join is internal)} \]

With \( r=5 \) cm and \( h=8 \) cm:

\[ A = 10\times8 + \frac{1}{2}\pi\times25 = 80 + 12.5\pi \approx 119.3 \text{ cm}^2 \] \[ P = 16 + 10 + 5\pi = 26 + 5\pi \approx 41.7 \text{ cm} \]

 Key Takeaways

  • \( C=2\pi r \), \( A=\pi r^2 \). Always check: radius or diameter given?
  • Composite area: sum of parts. Composite perimeter: only the outer boundary — not internal edges.
  • Semicircle perimeter contribution: arc only (\( \pi r \)), not the diameter.
  • Leave answers in terms of \( \pi \) unless a decimal is requested.