Lesson
Notes
Apothem Formula (Shortcut)
Apothem \( = \displaystyle \frac{s}{2 \tan{( \frac{180}{n}})}\)
s= side length of regular polygon
n= number of sides of regular polygon
Practice Problems
\(\textbf{1)}\) Find the area of the regular hexagon
The area \(=54\sqrt{3}\) and is approximately \( 93.53 \) in\(^2 \)
\(\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\text{Step 1: Find the apothem}\)
\(\,\,\,\,\,\,\tan(60)=\frac{a}{3}\)
\(\,\,\,\,\,\,a=3\tan(60)\)
\(\,\,\,\,\,\,a=3\sqrt{3}\)
\(\text{Step 2: Find the perimeter}\)
\(\,\,\,\,\,\,\text{Perimeter}=\text{(side length)} \cdot \text{number of sides}\)
\(\,\,\,\,\,\,\text{Perimeter}=6 \cdot 6 = 36\)
\(\text{Step 3: Calculate Area}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}3\sqrt{3} \cdot 36\)
\(\,\,\,\,\,\,\text{Area}=54\sqrt{3}\)
\(\,\,\,\,\,\,\text{Area} \approx 93.53 \) in\(^2 \)
\(\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\text{Step 1: Find the apothem}\)
\(\,\,\,\,\,\,\tan(60)=\frac{a}{3}\)
\(\,\,\,\,\,\,a=3\tan(60)\)
\(\,\,\,\,\,\,a=3\sqrt{3}\)
\(\text{Step 2: Find the perimeter}\)
\(\,\,\,\,\,\,\text{Perimeter}=\text{(side length)} \cdot \text{number of sides}\)
\(\,\,\,\,\,\,\text{Perimeter}=6 \cdot 6 = 36\)
\(\text{Step 3: Calculate Area}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}3\sqrt{3} \cdot 36\)
\(\,\,\,\,\,\,\text{Area}=54\sqrt{3}\)
\(\,\,\,\,\,\,\text{Area} \approx 93.53 \) in\(^2 \)
\(\textbf{2)}\) Find the area of the regular octagon
The area is approximately \( 173.82 \) in\(^2 \)
\(\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\text{Step 1: Find the apothem}\)
\(\,\,\,\,\,\,\tan(67.5)=\frac{a}{3}\)
\(\,\,\,\,\,\,a=3\tan(67.5)\)
\(\,\,\,\,\,\,a \approx 7.24264\)
\(\text{Step 2: Find the perimeter}\)
\(\,\,\,\,\,\,\text{Perimeter}=\text{(side length)} \cdot \text{number of sides}\)
\(\,\,\,\,\,\,\text{Perimeter}=8 \cdot 6 = 48\)
\(\text{Step 3: Calculate Area}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\,\,\,\,\,\,\text{Area} \approx \frac{1}{2}7.24264 \cdot 48\)
\(\,\,\,\,\,\,\text{Area} \approx 173.82\) in\(^2 \)
\(\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\text{Step 1: Find the apothem}\)
\(\,\,\,\,\,\,\tan(67.5)=\frac{a}{3}\)
\(\,\,\,\,\,\,a=3\tan(67.5)\)
\(\,\,\,\,\,\,a \approx 7.24264\)
\(\text{Step 2: Find the perimeter}\)
\(\,\,\,\,\,\,\text{Perimeter}=\text{(side length)} \cdot \text{number of sides}\)
\(\,\,\,\,\,\,\text{Perimeter}=8 \cdot 6 = 48\)
\(\text{Step 3: Calculate Area}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\,\,\,\,\,\,\text{Area} \approx \frac{1}{2}7.24264 \cdot 48\)
\(\,\,\,\,\,\,\text{Area} \approx 173.82\) in\(^2 \)
\(\textbf{3)}\) Find the area of the regular pentagon
The area is approximately \( 61.94 \) in\(^2 \)
\(\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\text{Step 1: Find the apothem}\)
\(\,\,\,\,\,\,\tan(54)=\frac{a}{3}\)
\(\,\,\,\,\,\,a=3\tan(54)\)
\(\,\,\,\,\,\,a \approx 4.12915\)
\(\text{Step 2: Find the perimeter}\)
\(\,\,\,\,\,\,\text{Perimeter}=\text{(side length)} \cdot \text{number of sides}\)
\(\,\,\,\,\,\,\text{Perimeter}=5 \cdot 6 = 30\)
\(\text{Step 3: Calculate Area}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\,\,\,\,\,\,\text{Area} \approx \frac{1}{2}4.12915 \cdot 30\)
\(\,\,\,\,\,\,\text{Area} \approx 61.94\) in\(^2 \)
\(\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\text{Step 1: Find the apothem}\)
\(\,\,\,\,\,\,\tan(54)=\frac{a}{3}\)
\(\,\,\,\,\,\,a=3\tan(54)\)
\(\,\,\,\,\,\,a \approx 4.12915\)
\(\text{Step 2: Find the perimeter}\)
\(\,\,\,\,\,\,\text{Perimeter}=\text{(side length)} \cdot \text{number of sides}\)
\(\,\,\,\,\,\,\text{Perimeter}=5 \cdot 6 = 30\)
\(\text{Step 3: Calculate Area}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\,\,\,\,\,\,\text{Area} \approx \frac{1}{2}4.12915 \cdot 30\)
\(\,\,\,\,\,\,\text{Area} \approx 61.94\) in\(^2 \)
\(\textbf{4)}\) Find the area of the regular decagon
The area is approximately \( 276.99 \) in\(^2 \)
\(\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\text{Step 1: Find the apothem}\)
\(\,\,\,\,\,\,\tan(72)=\frac{a}{3}\)
\(\,\,\,\,\,\,a=3\tan(72)\)
\(\,\,\,\,\,\,a \approx 9.23305\)
\(\text{Step 2: Find the perimeter}\)
\(\,\,\,\,\,\,\text{Perimeter}=\text{(side length)} \cdot \text{number of sides}\)
\(\,\,\,\,\,\,\text{Perimeter}=10 \cdot 6 = 60\)
\(\text{Step 3: Calculate Area}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\,\,\,\,\,\,\text{Area} \approx \frac{1}{2}9.23305 \cdot 60\)
\(\,\,\,\,\,\,\text{Area} \approx 276.99\) in\(^2 \)
\(\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\text{Step 1: Find the apothem}\)
\(\,\,\,\,\,\,\tan(72)=\frac{a}{3}\)
\(\,\,\,\,\,\,a=3\tan(72)\)
\(\,\,\,\,\,\,a \approx 9.23305\)
\(\text{Step 2: Find the perimeter}\)
\(\,\,\,\,\,\,\text{Perimeter}=\text{(side length)} \cdot \text{number of sides}\)
\(\,\,\,\,\,\,\text{Perimeter}=10 \cdot 6 = 60\)
\(\text{Step 3: Calculate Area}\)
\(\,\,\,\,\,\,\text{Area}=\frac{1}{2}\text{apothem} \cdot \text{perimeter}\)
\(\,\,\,\,\,\,\text{Area} \approx \frac{1}{2}9.23305 \cdot 60\)
\(\,\,\,\,\,\,\text{Area} \approx 276.99\) in\(^2 \)
See Related Pages\(\)
\(\bullet\text{ Geometry Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Intro to Polygons}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Angles of Polygons}\)
\(\,\,\,\,\,\,\,\,\text{Sum}=(n-2)180^{\circ}…\)
\(\bullet\text{ Population Density}\)
\(\,\,\,\,\,\,\,\,\text{Population Density}=\frac{\text{Number of People}}{\text{Area}}…\)
In Summary
An apothem is a line segment that is drawn from the center of a regular polygon to the midpoint of one of its sides. The area of a regular polygon can be calculated by multiplying the apothem by the perimeter, and then dividing the result by two. The area of regular polygons and the concept of an apothem is typically taught in geometry class, which is often taken in high school. We learn about these because they are useful tools for solving geometry problems and understanding the properties of shapes.
Real world examples of Area of Regular Polygons (Apothem)
Architects and Engineers: Architects and engineers often use the area of regular polygons and the concept of an apothem to calculate the amount of material needed for construction projects.
Landscapers and Gardeners: Landscapers and gardeners may use the area of regular polygons and the concept of an apothem to plan out the layout of a garden or other outdoor space.
Biology: The area of regular polygons and the concept of an apothem can be used to study the patterns and symmetry found in nature.
Computer Graphics: The area of regular polygons and the concept of an apothem can be used to create 3D models and animations.