Despite a Pentagon only having one more side/edge than a square or rectangle.
Working the area of a regular Pentagon in comparison can be slightly more involved and complex.
Formula for How to Find Pentagon Area
If we have a regular Pentagon such as the one shown below.
The area can be established with a general formula for how to find Pentagon area.
To use this formula we need to know the values of a and b first.
This page shows where this formula for the area of a Pentagon comes from, and how to acquire the value of a.
Derive Formula for Pentagon Area
Apothem and Side Length
For the case of a regular Pentagon, all 5 exterior angles add up to 360°.
As we can see, the 5 exterior angles of a regular Pentagon are of equal size.
No matter how large or small the size of the Pentagon is.
The size of 1 exterior angle is given by. \frac{360}{5} = 72°
This value can help tell us what size an interior angle of the Pentagon is.
Because it’s the case that one side of a straight line is of size, 180°.
So the size of an interior angle can be given by. 180° − 72° = 108°
Triangles in a Pentagon:
Now if we keep the size of the interior angles to the side briefly.
A regular Pentagon can be separated into 5 triangles of equal size.
By creating straight lines drawn from the central point of the Pentagon, to the corners.
As all of the triangles inside the Pentagon are the same size, if we can establish the area of just 1 of the triangles.
We can then work out the area of a entire Pentagon, by the multiplication of one triangle area by 5.
To find the area of 1 of the inside triangles though, some more information is needed than what we have so far.
Each of the straight lines present from the central point of the Pentagon to the corners, do in fact cut each corner in half evenly.
Seeing as each interior angle is of size 108°.
The subsequent size of each angle on each side of the straight lines can be given by. \frac{108\degree}{2} = 54°.
The next thing to realize, is that each of the 5 triangles inside the Pentagon, can themselves be divided into 2 smaller right angle triangles.
Each one of thee 10 new smaller triangles inside the Pentagon, is a right angle triangle.
A right angle triangle where we know the size of all the interior angles.
So now lets’ see how all this can help with establishing the area of a regular Pentagon.
Find Pentagon Area Example
Below we have Pentagon where all the sides/edges are 4cm in length.
As a standard regular Pentagon, this Pentagon can be separated up into a number of smaller triangles as seen earlier.
The angles in each of the smaller right angle triangles, are always the same size, regardless of how large or small the Pentagon is.
However the length of the triangles sides can differ.
With the Pentagon we have here, the base of one of the smaller right angle triangles will be 2cm, half of an entire 4cm sized edge.
Triangles in Pentagon:
As the area of a triangle is given by, \frac{1}{2} × BASE × HEIGHT.
Here we only need to find the size of the height, which is the red line.
As we then will be able to work out the whole area of the larger triangle with base 4cm.
Then, this triangle area multiplied by 5, will give us the area of the whole Pentagon.
Looking at one of the right angle triangles, we can use some Trigonometry to establish the size of the red line, the height.
tan = \frac{\tt{opposite}}{\tt{adjacent}}
For the angle 54° here, the adjacent side is 2cm, and the height is the opposite side.
tan( 54° ) = \frac{\tt{height}}{\tt{2}} ( × 2 )
2 × tan(54°) = height , 2.75 = height
Now that we know the sizes of both the base and the height of the larger triangle, we are able to find out the area.
AREA = \frac{\bf{1}}{\bf{2}} × 4 × 2.75 = 5.5cm2
So for the area of the whole Pentagon, as there are 5 larger triangles inside, we just multiply 5.5cm2 by 5.
AREA OF PENTAGON = 5.5cm2 × 5 = 27.5cm2
General Case:
As it turns out, all we really needed to establish the area of one of the 5 triangles inside, was the size of the red height line, along with the length of a side/edge of the Pentagon.
This red line that’s been used for the height of the triangles. Coming from the central point of an edge/side, to the central point of the whole Pentagon, isn’t just the line that represents the height of the triangles.
It’s also a specific straight line that is called the apothem.
Considering what we’ve seen on this page.
Area of a Pentagon was: ( AREA of 1 triangle ) × 5
With:
AREA of 1 triangle = \frac{1}{2} × s × a,
So: ( \frac{1}{2} × s × a ) × 5 = \bf{\frac{5}{2}}sa
This is how the formula for how to find Pentagon area arises, which we can make use of as long as we know the values of a and s.