Composite 2D figures (combining shapes for area)
Composite Two-Dimensional Figures
Composite Two-Dimensional Figures is a topic in Two-Dimensional Measurement in the Texas Essential Knowledge and Skills (Geometry, §111.41). It is aligned to Standard G.11(B), which requires students to apply area formulas to composite two-dimensional figures to solve problems.
A composite figure is decomposed into familiar shapes whose areas are added or subtracted.
Theory
A composite figure is made of several simple shapes. Find its area by decomposing it:
- Add the areas of the parts when shapes are joined.
- Subtract the area of a cut-out (a hole or removed corner) from the whole.
The strategy:
How to find a composite area
- Break the figure into familiar shapes.
- Find each shape's area.
- Add joined parts or subtract cut-outs.
- Keep \(\pi\) exact if circles are involved.
Add the rectangle and triangle areas.
| \(\text{rectangle}\) | \(=\) | \(10\times 6=60\) |
| \(\text{triangle}\) | \(=\) | \(\dfrac12(10)(4)=20\) |
| \(\text{total}\) | \(=\) | \(60+20=80\) |
Subtract the circle from the rectangle.
| \(\text{rectangle}\) | \(=\) | \(12\times 8=96\) |
| \(\text{circle}\) | \(=\) | \(\pi(3)^2=9\pi\) |
| \(\text{remaining}\) | \(=\) | \(96-9\pi\) |
Subtract the removed corner.
| \(10\times 8-4\times 3\) | \(=\) | \(80-12=68\) |
Add the rectangle and the semicircle (radius \(5\)).
| \(\text{rectangle}\) | \(=\) | \(10\times 4=40\) |
| \(\text{semicircle}\) | \(=\) | \(\dfrac12\pi(5)^2=\dfrac{25\pi}{2}\) |
| \(\text{total}\) | \(=\) | \(40+\dfrac{25\pi}{2}\) |
Common pitfalls
Frequently asked questions
How do you find the area of a composite figure?
Break it into familiar shapes, find each area, and add (or subtract cut-outs).
When do you subtract areas?
When a region is removed — a hole or a cut-out — subtract it from the whole shape.
How do you handle a circle in a composite figure?
Use \(\pi r^2\) (or half for a semicircle) and keep \(\pi\) exact unless a decimal is requested.
What is the most common mistake with composite areas?
Mixing up whether to add or subtract, and double-counting shared regions.