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Geometry Two-dimensional measurement

Composite 2D figures (combining shapes for area)

20 practice questions 0 video lessons Theory + worked examples

Composite Two-Dimensional Figures

Common Core Geometry • Standard G-MG.1 • Two-Dimensional Measurement

Composite Two-Dimensional Figures is a topic in Two-Dimensional Measurement in the Common Core State Standards. It is aligned to Standard G-MG.1, which requires students to use geometric shapes and their measures to model objects and compute composite areas.

A composite figure is decomposed into familiar shapes whose areas are added or subtracted.

Common Core Geometry › Two-Dimensional Measurement › Composite Two-Dimensional Figures  —  Standard G-MG.1

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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.
Split into shapes you know — rectangles, triangles, circles — then combine.
Composite figure: add areas A composite figure is broken into familiar shapes whose areas are added. rectangle semicircle add the areas of the parts
Add the areas of the rectangle and semicircle.
Composite figure: subtract areas A shaded region is found by subtracting the area of a cut-out from the whole shape. subtract the hole from the whole
Subtract the circular hole from the rectangle.

The strategy:

\[\text{total area}=\sum(\text{part areas})\ \text{or}\ (\text{whole})-(\text{cut-out})\]
add the areas of the parts, or subtract a cut-out from the whole
Watch shared edges: don't double-count or leave gaps when splitting.

How to find a composite area

  1. Break the figure into familiar shapes.
  2. Find each shape's area.
  3. Add joined parts or subtract cut-outs.
  4. Keep \(\pi\) exact if circles are involved.
Example 1 — Add two shapes
A figure is a \(10\times 6\) rectangle with a triangle of base \(10\) and height \(4\) on top. Find the total area.
Solution

Add the rectangle and triangle areas.

\(\text{rectangle}\)\(=\)\(10\times 6=60\)
\(\text{triangle}\)\(=\)\(\dfrac12(10)(4)=20\)
\(\text{total}\)\(=\)\(60+20=80\)
the total area is 80 square units
Example 2 — Subtract a hole
A \(12\times 8\) rectangle has a circle of radius \(3\) removed. Find the remaining area (leave \(\pi\)).
Solution

Subtract the circle from the rectangle.

\(\text{rectangle}\)\(=\)\(12\times 8=96\)
\(\text{circle}\)\(=\)\(\pi(3)^2=9\pi\)
\(\text{remaining}\)\(=\)\(96-9\pi\)
the remaining area is 96 minus 9 pi
Example 3 — An L-shape
An L-shape is a \(10\times 8\) rectangle with a \(4\times 3\) corner removed. Find its area.
Solution

Subtract the removed corner.

\(10\times 8-4\times 3\)\(=\)\(80-12=68\)
the area is 68 square units
Example 4 — Rectangle plus semicircle
A \(10\times 4\) rectangle has a semicircle of diameter \(10\) on one end. Find the area (leave \(\pi\)).
Solution

Add the rectangle and the semicircle (radius \(5\)).

\(\text{rectangle}\)\(=\)\(10\times 4=40\)
\(\text{semicircle}\)\(=\)\(\dfrac12\pi(5)^2=\dfrac{25\pi}{2}\)
\(\text{total}\)\(=\)\(40+\dfrac{25\pi}{2}\)
the total is 40 plus 25 pi over 2

Common pitfalls

Decide add vs subtract carefully. Holes are subtracted; joined pieces are added.
Use the right dimensions for each part, including radius from diameter.
Don't double-count overlaps when splitting the figure.

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.

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