Perimeter and Area of Plane Figures

Objective: Add all outer side lengths.

Prerequisite skills: length units and basic polygon properties. A perimeter measures the boundary length. For a rectangle with length l and width w, opposite sides are equal and P=2l+2w. A circle’s boundary uses circumference, developed separately. Common error: multiplying l by w when asked for perimeter. Restrictions and interpretation: Perimeter uses linear units; area uses square units; a triangle height must be perpendicular to its base.

Objective: Use a perpendicular height with the correct shape factor.

A parallelogram area is base times perpendicular height, independent of slanted side length. A triangle with the same base and height has half that area. A trapezoid area is the average of its two parallel bases times perpendicular height. Common error: using a slanted side as height without a perpendicular condition. Restrictions and interpretation: Perimeter uses linear units; area uses square units; a triangle height must be perpendicular to its base.

Objective: Use the mean of parallel bases in an area calculation.

The parallel bases may have different lengths, so a single base times height is insufficient. Pairing two congruent trapezoids forms a parallelogram with base b₁+b₂, which explains the factor 1/2. Common error: adding the nonparallel side lengths into the area formula. Restrictions and interpretation: Perimeter uses linear units; area uses square units; a triangle height must be perpendicular to its base. 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.