Shaded area

Shaded area questions will generally involve one of two things: a geometric shape (or possibly one inside of the other) or an inequality graph.

Geometric Shading Problems

Geometric shading problems will focus on calculating area. Students will be asked to calculate a shaded area from a larger whole area. This will often involve subtracting one area from another.

The most common version of this is to have a square inside of a circle or vice versa. It may also involve a triangle inside a sector of a circle. The question would then ask for the area of the circle sector minus the area of the triangle.

Inequality Graphs

Inequality shading problems will focus on identifying areas on a graph. Students will need to identify which area above or below a line should be shaded, or which portions of a shape should be shaded to satisfy an inequality.

Like with normal inequality problems, the easiest way to approach these questions is to replace the inequality sign with an equal sign. Normal algebra rules then apply for the equation: isolate the y on the left hand side to make it look like the usual y = mx + b form.

Here is an example problem from the June 2017 ACT:

ax + by = c can be rewritten as 

y= −axb+cb. Remembering that 0 < a < b < c. The slope of the line will be negative, so the line should be downward sloping. This eliminates F. 

cbis positive, so the y-intercept should be positive: this eliminates J and H.

The only difference between G and K is that one has shading below the line, while the other has shading above the line.

• Shading above a line means that y should be greater than the other side of the equation
• Shading below a line means that y should be less than the other side of the equation

The inequality version of the equation is: 

y≤ −axb+cb. Y is less than the other side of the equation, meaning that the shading should be below the line.

This means the answer is K.