Find the perimeter and area of polygons in the coordinate plane using the distance formula and area formulas, and apply these to modeling problems (A.GSR and A.MM, Geometric and Spatial Reasoning and Modeling).

What this topic is asking

This standard from Geometric and Spatial Reasoning (A.GSR) and Mathematical Modeling (A.MM) asks you to find the perimeter and area of polygons drawn on the coordinate plane, combining the distance formula with the familiar area formulas. The Georgia Milestones EOC tests this with numeric-entry items (find perimeter and area from vertices) and constructed-response modeling items (a plot of land, a garden, a sign). The work reuses the distance formula from this module and the area and perimeter formulas, several of which are typically on the reference sheet.

Side lengths on the coordinate plane

Each side of a polygon is a segment, so its length is the distance between its endpoints.

• For a side parallel to an axis, the length is simply the difference of the changing coordinate: the segment from $(1, 1)$ to $(5, 1)$ has length $5 - 1 = 4$ (only $x$ changes).
• For a slanted side, use the full distance formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Perimeter

The perimeter is the total distance around the figure, the sum of all side lengths. For a rectangle with width $w$ and height $h$, perimeter $= 2(w + h)$.

Area formulas

Use the formula that fits the figure.

For a right triangle with legs along the axes, the two legs are the base and height. A triangle with vertices $(0, 0), (6, 0), (0, 4)$ has base 6, height 4, area $\frac{1}{2}(6)(4) = 12$ square units.

How the Milestones examines this topic

• Numeric entry. Find the perimeter or area of a polygon given its vertices.
• Multiple choice. Choose the area or perimeter, with unit (linear versus square) distractors.
• Constructed response. Solve a modeling problem (fencing, flooring, signage) using perimeter or area, with units.

Why axis-parallel sides are easy

A useful efficiency on the EOC is recognizing when you do not need the full distance formula. When a side is horizontal, both endpoints share the same $y$-coordinate, so the distance formula's $(y_2 - y_1)^2$ term is zero and the length collapses to $\sqrt{(x_2 - x_1)^2} = |x_2 - x_1|$, just the difference of the $x$-coordinates. The same holds for a vertical side, where the length is $|y_2 - y_1|$. So a rectangle aligned with the axes needs only two subtractions, not two distance-formula computations. Spotting which sides are axis-parallel before reaching for the formula saves time and reduces arithmetic errors, and it is why coordinate figures are often drawn aligned with the axes on the test.

Keeping perimeter and area units straight

A frequent point-loser is confusing the units of perimeter and area, so it is worth fixing the idea. Perimeter is a total distance around the figure, measured in the same linear units as the sides (feet, meters), and it is a single length. Area is the amount of surface enclosed, measured in square units (square feet, square meters), because it comes from multiplying two lengths. So a question asking how much fencing surrounds a garden wants the perimeter (linear units), while how much sod covers it wants the area (square units). Reading the context for "around" versus "covers" and attaching the correct units, linear for perimeter and square for area, is part of the modeling credit on constructed-response items.

Try this

Q1. Find the perimeter of a square with vertices $(0, 0), (3, 0), (3, 3), (0, 3)$. [1 point]

• Cue. Side 3, perimeter $4 \times 3 = 12$ units.

Q2. Find the area of a right triangle with vertices $(0, 0), (8, 0), (0, 5)$. [1 point]

• Cue. $\frac{1}{2}(8)(5) = 20$ square units.