Calculate the Area of an Equilateral Hexagon with Side Length 7 and Height 12.124

Hexagon Area with Trapezoid Decomposition

Below is an equilateral hexagon.

AB = 7
FC = 14
AE = 12.124

77712.12412.12412.124141414AAABBBCCCDDDEEEFFFGGG

What is the area of the hexagon?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:12 Let's find the area of this hexagon together!
00:15 We'll use a special formula for the area of a regular hexagon.
00:23 Check the given data for the side length.
00:31 This hexagon is equilateral, meaning all sides are equal.
00:38 Now, let's plug in the side length into our formula and find the area!
00:59 And there you have it! That's how we solve this problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Below is an equilateral hexagon.

AB = 7
FC = 14
AE = 12.124

77712.12412.12412.124141414AAABBBCCCDDDEEEFFFGGG

What is the area of the hexagon?

2

Step-by-step solution

The hexagon consists of two equal trapezoids, so we will strive to calculate the area of one of them and multiply it.

AFE is an isosceles triangle,

its height (FG) crosses the base exactly in the center, therefore:

AG=GE AG=GE

AG=12AE AG=\frac{1}{2}AE

We replace and discover:

AE=12×12=6 AE=\frac{1}{2}\times12=6

We replace the data in the formula for the area of a trapezoid:

(base+base)2×altura \frac{(base+base)}{2}\times altura

7+142×6=212×6=10.5×6=63 \frac{7+14}{2}\times6=\frac{21}{2}\times6=10.5\times6=63

63 is the area of half of the hexagon, therefore:

63×2=126 63\times2=126

3

Final Answer

127.3

Key Points to Remember

Essential concepts to master this topic
  • Decomposition: Regular hexagon equals two identical trapezoids sharing same base
  • Technique: Use trapezoid formula (b1+b2)2×h \frac{(b_1+b_2)}{2} \times h with bases 7 and 14
  • Check: Both trapezoids have same area: 63.65 × 2 = 127.3 ✓

Common Mistakes

Avoid these frequent errors
  • Using wrong height measurement
    Don't use AE = 12.124 as trapezoid height = wrong calculation! AE is the hexagon's full height, but trapezoid height is half that distance. Always use AG = GE = 6.062 as the perpendicular height for each trapezoid.

Practice Quiz

Test your knowledge with interactive questions

Given the following trapezoid:

AAABBBCCCDDD584

Calculate the area of the trapezoid ABCD.

FAQ

Everything you need to know about this question

Why does the hexagon split into two trapezoids?

+

A regular hexagon has a vertical line of symmetry through its center. This line divides it into two identical trapezoids that share the same dimensions, making calculation easier!

How do I find the height of each trapezoid?

+

The trapezoid height is the perpendicular distance from one parallel base to the other. Since AE = 12.124 is the full hexagon height, each trapezoid has height = 12.1242=6.062 \frac{12.124}{2} = 6.062 .

What are the parallel bases of the trapezoid?

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The two parallel sides are AB = 7 (top base) and the middle section FC = 14 (bottom base). These form the parallel edges of each trapezoid.

Why do I multiply by 2 at the end?

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You calculated the area of one trapezoid, but the hexagon contains two identical trapezoids. So: Total Area = 2 × (One Trapezoid Area).

Can I use the regular hexagon area formula instead?

+

The standard formula 332s2 \frac{3\sqrt{3}}{2}s^2 only works for regular hexagons where all sides are equal. This hexagon has different side lengths, so use the trapezoid method!

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