Enclosure Method Answer Key
About This Worksheet
This worksheet helps tenth-grade students find the area of polygons by placing each figure inside the smallest coordinate-aligned rectangle and then subtracting the extra space around it. Students work with four figures, including a rotated rectangle, triangle, irregular quadrilateral, and rotated pentagon. Instead of breaking the figure apart into smaller pieces, they begin with a larger rectangle that completely surrounds it and remove the outside regions. This gives students a second useful strategy for finding area on the coordinate plane.
For example, students first identify the leftmost, rightmost, highest, and lowest coordinates to determine the dimensions of the enclosing rectangle. They calculate that rectangle’s area, find the areas of the extra triangles or smaller regions outside the polygon, and subtract those pieces from the rectangle. This helps students see that area can be found by subtraction just as effectively as by decomposition.
What Students Practice
Students practice identifying the smallest horizontal-and-vertical rectangle that encloses a coordinate polygon. They then calculate the rectangle’s area and determine which outside pieces must be removed. This requires careful use of coordinate differences, triangle area, and basic subtraction.
The worksheet also helps students compare different area strategies. Some figures are easier to solve by decomposition, while others are easier to enclose and subtract. Learning to recognize that difference gives students more flexibility.
Why This Skill Matters
Not every coordinate polygon is easy to divide into familiar shapes. An enclosing rectangle can make a difficult-looking figure much easier to manage. Students learn that there is often more than one good method for finding area.
This strategy is especially useful with rotated figures and irregular polygons. It encourages students to think about the space around a figure, not just the space inside it. That kind of visual reasoning is valuable throughout geometry.
Common Challenges
Students may choose an enclosing rectangle that is larger than necessary. Remind them to use the extreme x- and y-values so the rectangle fits as tightly around the figure as possible. A smaller enclosure usually makes the subtraction easier.
Another common mistake is forgetting one of the outside regions. Encourage students to shade or label every section that is not part of the polygon before subtracting. This helps make sure the extra area is removed exactly once.
How To Use It
Teachers can use this worksheet after students have practiced decomposition as another coordinate-area method. It works well for guided practice, partner discussion, homework, or a strategy-comparison lesson. Showing the same polygon solved both ways can help students see why one method may be more efficient.
Parents and homeschool teachers can ask, “What is the smallest rectangle that would hold this whole shape?” Then have the student find its area and look for the pieces outside the polygon that need to be removed. This turns a complicated shape into a simpler subtraction problem.
Grade Alignment
This worksheet is designed for Grade 10 geometry and supports CCSS HSG-GPE.B.7, which includes using coordinates to compute areas of polygons. It also reinforces decomposition and subtraction strategies for composite figures. The enclosing method gives students another way to reason about irregular coordinate shapes.
Students should already know basic polygon area formulas and coordinate differences. After this practice, they can compare methods and choose the most efficient strategy for a given figure.