Calculate Hexagon Area with 28 cm² Rectangle Inside: Geometry Problem

Q: Why can't I just use the rectangle area as the hexagon area?

The rectangle is inside the hexagon, not the same size! Think of it like a pizza slice inside a whole pizza - they're different sizes. You need to use the rectangle to find the hexagon's side length first.

Q: How do I know the hexagon has equal sides?

The problem states it's a regular hexagon, which means all sides are equal. This is why finding one side length (4 cm) tells us all sides are 4 cm.

Q: What's the hexagon area formula and why is it so complicated?

The formula $ \frac{6 \times a^2 \times \sqrt{3}}{4} $ comes from dividing a hexagon into 6 triangles. Don't worry about memorizing it - focus on understanding that you need the side length first!

Q: Why do I get a decimal answer when the rectangle area was a whole number?

This is normal! The hexagon area involves $ \sqrt{3} $ which is irrational, so you get approximately 41.56 cm². The rectangle having a whole number area doesn't mean the hexagon will too.

Q: How can I check if my hexagon area is reasonable?

Compare it to the rectangle area! Since the hexagon contains the rectangle, its area (41.56 cm²) should be larger than the rectangle's area (28 cm²). This makes sense!

Hexagon Area with Rectangle Dimensions

Below is a hexagon that contains a rectangle inside it.

The area of the rectangle is 28 cm².

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

00:06 Let's find the area of this hexagon.

00:09 We use the formula for a rectangle's area by multiplying side by side.

00:14 Next, we substitute the values given to solve for A.

00:19 Now, let's isolate A in the equation.

00:22 This side of the rectangle is also the side of the regular hexagon.

00:27 Remember, in a regular hexagon, all sides are equal.

00:31 We'll apply the formula for a regular hexagon's area.

00:40 Substitute the side value and find the area.

00:48 Let's simplify our calculations.

00:54 And there we have it, our solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Below is a hexagon that contains a rectangle inside it.

The area of the rectangle is 28 cm².

What is the area of the hexagon?

Step-by-step solution

Since we are given the area of the rectangle, let's first work out the length of the missing side:

$7\times a=28$

We'll now divide both sides by 7 to get:

$a=4$

Since all sides are equal in a hexagon, each side is equal to 4.

Now let's calculate the area of the hexagon:

$\frac{6\times a^2\times\sqrt{3}}{4}$

$\frac{6\times4^2\times\sqrt{3}}{4}$

Finally, we simplify the exponent in the denominator of the fraction to get:

$6\times4\times\sqrt{3}=24\times\sqrt{3}=41.56$

Final Answer

41.56

Key Points to Remember

Essential concepts to master this topic

Rectangle Formula: Use given area to find missing side dimension
Technique: From area 28 and height 7, calculate width: 28 ÷ 7 = 4
Check: Substitute into hexagon formula: $\frac{6 \times 4^2 \times \sqrt{3}}{4} = 24\sqrt{3} \approx 41.56$ ✓

Common Mistakes

Avoid these frequent errors

Using rectangle area as hexagon area
Don't assume the hexagon area equals the rectangle area of 28 cm² = completely wrong answer! The rectangle is just part of the hexagon, not the whole shape. Always find the hexagon's side length first, then use the proper hexagon area formula.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle ABCD below.

Side AB is 6 cm long and side BC is 4 cm long.

What is the area of the rectangle?

FAQ

Everything you need to know about this question

Why can't I just use the rectangle area as the hexagon area?

The rectangle is inside the hexagon, not the same size! Think of it like a pizza slice inside a whole pizza - they're different sizes. You need to use the rectangle to find the hexagon's side length first.

How do I know the hexagon has equal sides?

The problem states it's a regular hexagon, which means all sides are equal. This is why finding one side length (4 cm) tells us all sides are 4 cm.

What's the hexagon area formula and why is it so complicated?

The formula $\frac{6 \times a^2 \times \sqrt{3}}{4}$ comes from dividing a hexagon into 6 triangles. Don't worry about memorizing it - focus on understanding that you need the side length first!

Why do I get a decimal answer when the rectangle area was a whole number?

This is normal! The hexagon area involves $\sqrt{3}$ which is irrational, so you get approximately 41.56 cm². The rectangle having a whole number area doesn't mean the hexagon will too.

How can I check if my hexagon area is reasonable?

Compare it to the rectangle area! Since the hexagon contains the rectangle, its area (41.56 cm²) should be larger than the rectangle's area (28 cm²). This makes sense!

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