Composite Figures and Missing Dimensions

Objective: Find composite area by subtracting a hole from a whole.

Prerequisite skills: rectangle and triangle areas and reading textual dimensions. If a smaller region lies fully inside a larger one and is removed, the remaining area equals outer area minus removed area. Compute both in the same square units. Common error: subtracting lengths rather than areas. Restrictions and interpretation: Subtract only regions that lie inside the stated whole; do not assume a figure is drawn to scale.

Objective: Add nonoverlapping pieces to find a total.

A region can be split into rectangles and triangles whose interiors do not overlap. Add their areas once each. Shared boundaries have zero area and therefore do not need an area correction. Common error: adding a surrounding bounding rectangle that includes empty space. Restrictions and interpretation: Subtract only regions that lie inside the stated whole; do not assume a figure is drawn to scale.

Objective: Use a composite-area equation to recover a length.

Treat an unknown dimension as a variable in the area formula. Solve after subtracting any known component area. Check the resulting length is positive and fits the stated arrangement. Common error: dividing before subtracting the known extra area. Restrictions and interpretation: Subtract only regions that lie inside the stated whole; do not assume a figure is drawn to scale. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.