Calculate the Area of a Regular Hexagon with Length 4

Q: Why is the hexagon formula so complicated with square root of 3?

The formula comes from dividing a regular hexagon into 6 equilateral triangles. Each triangle has area $ \frac{\sqrt{3}}{4}s^2 $, so 6 triangles give us $ \frac{6\sqrt{3}}{4}s^2 = \frac{3\sqrt{3}}{2}s^2 $.

Q: Do I need to memorize the exact value of square root of 3?

No! You can always use the approximation $ \sqrt{3} \approx 1.732 $ for calculations. Just remember this common approximation and you'll be fine.

Q: How do I know which measurement in the diagram is the side length?

Look for a measurement along one of the 6 equal sides of the hexagon. In this problem, the blue segment labeled "4" represents the side length of the regular hexagon.

Q: What if I calculated 96 or 384 as my answer?

You likely used incorrect formulas! 96 might come from $ 6 \times 4^2 $ (wrong), and 384 from $ 24 \times 4^2 $ (also wrong). Always use the correct hexagon formula.

Q: Can I split the hexagon into rectangles instead of triangles?

While possible, it's much more complicated! The triangle method is easier because regular hexagons naturally divide into 6 congruent equilateral triangles from the center.

Regular Hexagon Area with Side Length

Given the hexagon in the drawing:

What is the area?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the area of the regular hexagon

00:03 Draw an equilateral triangle from the center of the hexagon

00:07 In an equilateral triangle all sides are equal

00:11 We'll use the formula for calculating the area of a regular hexagon

00:17 The side length according to the triangle we drew

00:25 We'll substitute the side length and solve for the area

00:31 We'll break down 4 squared into 4 times 4 and reduce

00:45 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the hexagon in the drawing:

What is the area?

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the given information in the drawing.
Step 2: Apply the formula for the area of a regular hexagon.
Step 3: Perform the necessary calculations using available data.

Now, let's work through each step:

Step 1: From the drawing, a side length of "4" is provided, indicated next to a blue segment. Assuming this corresponds to the side length of the regular hexagon.

Step 2: We'll use the formula for the area of a regular hexagon, which is $\frac{3\sqrt{3}}{2} \times s^2$ .

Step 3: Plugging in the side length $s = 4$ , our calculation is:

\text{Area} = \frac{3\sqrt{3}}{2} \times (4)^2 = \frac{3\sqrt{3}}{2} \times 16 = 24\sqrt{3}

Approximating $\sqrt{3} \approx 1.732$ , we have:

\text{Area} \approx 24 \times 1.732 = 41.568

Rounding this value gives approximately 41.56, which matches the given correct answer choice $41.56$ .

Final Answer

41.56

Key Points to Remember

Essential concepts to master this topic

Formula: Regular hexagon area equals $\frac{3\sqrt{3}}{2} \times s^2$
Calculation: With s = 4, area = $\frac{3\sqrt{3}}{2} \times 16 = 24\sqrt{3}$
Check: Approximate $\sqrt{3} \approx 1.732$ , so $24 \times 1.732 = 41.568$ ✓

Common Mistakes

Avoid these frequent errors

Using wrong formula or confusing with other polygon formulas
Don't use triangle area formula or square area formula = completely wrong answer! A hexagon has 6 sides and requires its specific formula. Always use $\frac{3\sqrt{3}}{2} \times s^2$ for regular hexagon area.

Practice Quiz

Test your knowledge with interactive questions

A hexagon has sides measuring $$ 8 $$ cm long. What is the area of the hexagon?

FAQ

Everything you need to know about this question

Why is the hexagon formula so complicated with square root of 3?

The formula comes from dividing a regular hexagon into 6 equilateral triangles. Each triangle has area $\frac{\sqrt{3}}{4}s^2$ , so 6 triangles give us $\frac{6\sqrt{3}}{4}s^2 = \frac{3\sqrt{3}}{2}s^2$ .

Do I need to memorize the exact value of square root of 3?

No! You can always use the approximation $\sqrt{3} \approx 1.732$ for calculations. Just remember this common approximation and you'll be fine.

How do I know which measurement in the diagram is the side length?

Look for a measurement along one of the 6 equal sides of the hexagon. In this problem, the blue segment labeled "4" represents the side length of the regular hexagon.

What if I calculated 96 or 384 as my answer?

You likely used incorrect formulas! 96 might come from $6 \times 4^2$ (wrong), and 384 from $24 \times 4^2$ (also wrong). Always use the correct hexagon formula.

Can I split the hexagon into rectangles instead of triangles?

While possible, it's much more complicated! The triangle method is easier because regular hexagons naturally divide into 6 congruent equilateral triangles from the center.

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