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Regular Polygon Area Explorer

Run the Regular Polygon Area Explorer MicroSim Fullscreen
Edit in the p5.js Editor

About This MicroSim

A regular polygon can be cut into congruent triangles that meet at its center. Each triangle has a base equal to one side \(s\) and a height equal to the apothem \(a\), the distance from the center to the midpoint of a side. Adding the \(n\) triangles gives

\[A = n \cdot \tfrac{1}{2} s a = \tfrac{1}{2} a P\]

where \(P = ns\) is the perimeter. The MicroSim draws the apothem (dashed red), the radius (blue), and one shaded triangle, and substitutes the current values into the formula.

The last panel compares the polygon with the circle drawn through its vertices. As the number of sides grows, the polygon fills more and more of that circle.

How to Use

  1. Drag Number of sides from 3 to 12 and watch the polygon's name, side length, apothem, perimeter, and area change.
  2. Drag Radius to make the polygon larger or smaller without changing its shape.
  3. Turn Show triangle subdivision off and on to see the n congruent triangles the formula is built from.
  4. Watch the Approaching a Circle panel as the number of sides increases.

Iframe Embed Code

<iframe src="main.html"
        height="557"
        width="100%"
        scrolling="no"></iframe>

Lesson Plan

Grade Level

9-12 (High School Geometry)

Duration

10-15 minutes

Prerequisites

Area of a triangle, perimeter, and the definition of a regular polygon.

Bloom's Taxonomy Level

Apply (L3)

Learning Objective

Students will be able to calculate the area of a regular polygon with A = ½aP and analyze how the apothem, the perimeter, and the number of sides affect the area.

Activities

  1. Derive (4 min): With the subdivision showing, students explain in their own words why n triangles of base s and height a give A = ½aP.
  2. Calculate (5 min): Students set n = 6 and a radius of their choice, compute the area by hand from the displayed a and P, and compare.
  3. Analyze (5 min): Keeping the radius fixed, students record the percentage of the circle filled for n = 3, 4, 6, 8, and 12 and describe the trend.

Assessment

Given a regular octagon with apothem 6.93 units and side length 5.74 units, students compute the perimeter and the area, then predict whether a regular 12-gon with the same radius has a larger or smaller area and justify the prediction.

References

  1. Wikipedia. Regular polygon.
  2. Wikipedia. Apothem.
  3. Common Core State Standards Initiative. High School Geometry: Geometric Measurement and Dimension (HSG-GMD.A.1).