How to Find the Area of a Pentagon
Start with the side length and apothem., Divide the pentagon into five triangles., Calculate the area of a triangle., Multiply by five to find the total area.
Step-by-Step Guide
-
Step 1: Start with the side length and apothem.
This method works for regular pentagons, with five equal sides.
Besides the side length, you'll need the "apothem" of the pentagon.
The apothem is the line from the center of the pentagon to a side, intersecting the side at a 90º right angle.
Don't confuse the apothem with the radius, which touches a corner (vertex) instead of a midpoint.
If you only know the side length and radius, skip down to the next method instead.
We'll use an example pentagon with side length 3 units and apothem 2 units. -
Step 2: Divide the pentagon into five triangles.
Draw five lines from the center of the pentagon, leading to each vertex (corner).
You now have five triangles. , Each triangle has a base equal to the side of the pentagon.
It also has a height equal to the pentagon's apothem. (Remember, the height of a triangle runs from a vertex to the opposite side, at a right angle.) To find the area of any triangle, just calculate ½ x base x height.
In our example, area of triangle = ½ x 3 x 2 = 3 square units. , We've divided the pentagon into five equal triangles.
To find the total area, just multiply the area of one triangle by five.
In our example, A(total pentagon) = 5 x A(triangle) = 5 x 3 = 15 square units. -
Step 3: Calculate the area of a triangle.
-
Step 4: Multiply by five to find the total area.
Detailed Guide
This method works for regular pentagons, with five equal sides.
Besides the side length, you'll need the "apothem" of the pentagon.
The apothem is the line from the center of the pentagon to a side, intersecting the side at a 90º right angle.
Don't confuse the apothem with the radius, which touches a corner (vertex) instead of a midpoint.
If you only know the side length and radius, skip down to the next method instead.
We'll use an example pentagon with side length 3 units and apothem 2 units.
Draw five lines from the center of the pentagon, leading to each vertex (corner).
You now have five triangles. , Each triangle has a base equal to the side of the pentagon.
It also has a height equal to the pentagon's apothem. (Remember, the height of a triangle runs from a vertex to the opposite side, at a right angle.) To find the area of any triangle, just calculate ½ x base x height.
In our example, area of triangle = ½ x 3 x 2 = 3 square units. , We've divided the pentagon into five equal triangles.
To find the total area, just multiply the area of one triangle by five.
In our example, A(total pentagon) = 5 x A(triangle) = 5 x 3 = 15 square units.
About the Author
Jeffrey Powell
Enthusiastic about teaching hobbies techniques through clear, step-by-step guides.
Rate This Guide
How helpful was this guide? Click to rate: