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Course: Geometry (all content)>Unit 13

Lesson 1: Special right triangles

Area of a regular hexagon

Using what we know about triangles to find the area of a regular hexagon. Created by Sal Khan.

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• What is the formula of a hexagon?
• The formula to calculate the area of a regular hexagon with side length s:
(3 √3 s^2)/2
Remember, this only works for REGULAR hexagons. For irregular hexagons, you can break the parts up and find the sum of the areas, depending on the shape.
Hope that helped!
• instead of dividing the hexagon into 6 triangles wouldn't it be slightly easier to draw a hypothetical line from point f to point b and again from point e to point c turning it into 2 triangles and a rectangle?
• You can try it and see. The advantage to dividing the hexagon into six congruent triangles is that you only have to calculate the area of one shape (and then multiply that answer by 6) instead of needing to find the area of both a rectangle and a triangle.
• At , isn't the area of an equilateral triangle (sqrt(3)*s^2)/4? I still get 3*sqrt(3), so I guess it's not as important as I thought...
• Well, you are actually right. That would be the special formula that gives you the area of equilateral triangles.
However the general area formula for triangles used in the video (A = 1/2*h*b), works for all triangles, including equilateral ones.
Using the special formula as suggested by you would have been quicker though, as you only need to know the side measurement of the equilateral, while the general formula requires the height and the base measurement.
• Couldn't you just divide it into separate triangles and add up the area of those?
• Yes, however formulas save time. This shape is small, but what about if it had 100 sides? Do you really want to calculate that many triangles. And each one of those triangles, you would need both the base and the height, which might not be given.
• What is the length of a side of a regular six sided polygon with radius of 8cm?
• Radius is the distance from the center to a corner. it is also important to know the apothem This works for any regular polygon.

Choose a side and form a triangle with the two radii that are at either corner of said side. You know both radii are 8 cm, which means you have an isosceles triangle.

You want to count how many of these triangles you can make. Basically each side will have one of these. this means each triangle will have an angle of measure 360/n, where n is the number of sides. In your case that is 360/6 =60. Since it is a scalene triangle you know the measure of the other two angles are the same. Also, you should know the angles of a triangle add up to 180. so in other words use some algebra to find the two other angles. Here that works out like this.

one angle is 60 and the other two are some other angle x where all three equal 180. So that works out to 60 + x + x = 180.
60 + 2x = 180
2x = 120
x = 60.

So this shows al four angles are 60 degrees, which means not only is it a scalene triangle, but an equilateral triangle. This means all sides are the same. And since we know the radii that means the remaining side is the sme measure at 8 cm.

If these were not equilateral you would have to use the apothem and the Pythagorean theorem.
• i dont get this. whats going on?
• Sal is explaining how to get the area of a regular hexagon. Try rewatching the video if you don't understand.
• Is there a video or tutorial on how to find the area of the hexagon in the case when it is inscribed into a circle?
• This is something one can notice when drawing silly shapes with compass;

The radius of the circle is equal to the side length of the hexagon.

Knowing that, you should be able to do it with this video : )
• what about a polygon? I'm confused.
• A hexagon is a polygon as are squares, triangles, rectangles, octagons and many other shapes. For each shape the formula to find the area will be different.
• At you failed to mention that all exterior angles are congruent and have the same measure as well as the interior angles. Of course, even if the hexagon isn't regular and all sides aren't congruent, the exterior angles could still be congruent provided they are attached the right kind of polygon. Anyways, I just felt like pointing that out because it really itched my brain. Side note: Thanks for the great math videos, they really help!
• I feel like defending Khan here, and I don't want to be a jerk, but:

He doesn't need to point out that the exterior angles are congruent because it's not relevant to what he's trying to solve: the area of the hexagon. Why mention it if it could be confusing the audience of why it's important?

He also told us that the angles all have the same measure at , which also means the interior angles are congruent, as by the Definition of Congruent Angles.

Your second argument was confusing, yet I get what you mean. Imagine that AB and DE were 4 units long, which would keep the interior angles at 120 degrees and thus the exterior angles congruent. Yet, again, the argument is about exterior angles, and exterior angles are not needed to find the area.