Shape Swap
About This Worksheet
This worksheet helps tenth-grade students explore how two figures can have the same area but different perimeters. Students first find the area and perimeter of each original coordinate-plane figure. Then they create a new rectangle or pentagon with the same area but a different perimeter, list the new vertices, and verify both measurements. This turns the activity into a design challenge instead of a routine calculation page.
For example, two rectangles can cover the same number of square units even if one is long and narrow while the other is closer to a square. Their areas can match while the total distance around them is different. Students use this idea to create their own coordinate figures that meet the required conditions.
What Students Practice
Students practice calculating area and perimeter from coordinates and then using those measurements to create a new figure. They must choose side lengths or vertices carefully so the area stays the same while the perimeter changes. This reinforces the fact that area and perimeter measure different features of a shape.
The worksheet also develops coordinate design skills. Students must list vertices that actually create the intended figure and then check their work mathematically. This gives them practice moving from calculation into construction.
Why This Skill Matters
Students sometimes assume that figures with the same area must also have the same perimeter. This worksheet directly challenges that misconception. By building examples themselves, students see that one measurement does not determine the other.
The skill also strengthens geometric reasoning and design thinking. Students must work backward from a condition and create a figure that satisfies it. That is a more advanced kind of problem solving than simply applying a formula.
Common Challenges
Students may accidentally create a new figure with both the same area and the same perimeter. Remind them that they need to check both conditions separately. The area must stay equal, but the perimeter must change.
Another common mistake is listing vertices that do not create the intended shape in the correct order. Encourage students to sketch the figure after choosing their coordinates and connect the vertices in sequence. This makes it easier to catch placement errors.
How To Use It
Teachers can use this worksheet as an extension activity after students have practiced standard area and perimeter problems. It works well for independent work, partner challenges, or a geometry design station. Comparing different student-created answers can also show that many solutions are possible.
Parents and homeschool teachers can ask, “Can you make the same amount of space with a different shape?” Then have the student choose new dimensions, place the vertices, and check both area and perimeter. This keeps the activity creative while still requiring exact math.
Grade Alignment
This worksheet is intended for Grade 10 geometry and supports CCSS HSG-GPE.B.7 through coordinate-based area and perimeter calculations. It also encourages construction, reasoning, and comparison of geometric properties. The open-ended design component adds a deeper level of understanding.
Students should already know how to find area and perimeter from coordinates. After this practice, they can move into comparison tasks, optimization ideas, and more complex coordinate design problems.