ACT® · Math: Geometry — Geometric Measurement and Modeling
Composite Areas and Shaded Regions
A shaded or composite area is almost always a big shape's area minus a smaller shape's area, or several simple shapes added together. Break the figure into rectangles, triangles, and circle pieces you can actually compute, then add or subtract.
Most shaded-region problems look scary because the picture is one weird figure, but they're built from shapes you already know. The trick is to see the whole figure as a combination of a few basic ones: rectangles, triangles, circles, semicircles. If part of the figure is cut out or sits on top of another shape, the shaded area is usually the outer shape's area minus the inner one's. If the figure is made of separate pieces placed side by side, you add their areas instead.
The mechanism is always the same three steps. First, identify the simple shapes hiding inside the composite one — draw lines if it helps split a weird figure into a rectangle plus a triangle, or a rectangle plus two semicircles. Second, find each piece's area using its own formula. Third, combine: subtract areas that are removed or overlapping, add areas that are just placed next to each other with no overlap. A rectangle with a circular hole is area of rectangle minus area of circle. A track shape made of a rectangle with two semicircular caps is the rectangle's area plus the two semicircles, which together make one full circle.
This connects directly to surface area and volume problems, which are really the same subtraction-and-addition idea in three dimensions — a box with a cylindrical hole drilled through it, for instance. It also feeds geometric probability, where you compare a shaded sub-region's area to the whole figure's area to get a probability. Get comfortable identifying hidden shapes here, and those later topics become the same skill with one extra layer on top.
Key terms
- Composite figure
- A shape made of two or more simple shapes joined together or with pieces removed.
- Shaded region
- The leftover area in a figure after one shape has been cut out of or placed over another.
- Overlap
- The part shared by two regions, which gets counted twice if you simply add their areas.
- Semicircle
- Half of a circle, with area equal to half of π r squared.
Watch it explained
- How it worksHow do you break a composite figure into rectangles, triangles and circle pieces you already know?
- Worked exampleFind the shaded area when a circle of radius 3 is cut from the middle of a 10 by 8 rectangle.
- Worked exampleFind the area of a running track made from a 60 by 40 rectangle capped by two semicircles of diameter 40.
- Common mistakeWhy doesn't adding the areas of two overlapping pieces give the area of the region they cover?
- Going furtherWhat happens to the shaded area as the removed circle grows until it touches both long sides?
Check yourself
End of topic.