Calculate Parallelogram Area: Rectangle with Perimeter 24 Inside

Q: Why can't I use BC = 5 as the base for the parallelogram area?

BC = 5 is a slanted side, not perpendicular to any height we can easily find. For area calculations, you need a base that's perpendicular to the height. Use AB = 11 as the base with height EC = 4.

Q: What if I calculated the rectangle's dimensions wrong?

Double-check using the perimeter formula: $ 2(8 + 4) = 24 $ ✓. If AE = 8 and perimeter = 24, then EC must equal 4. Always verify your rectangle dimensions add up correctly!

Parallelogram Area with Rectangle Perimeter

The parallelogram ABCD contains the rectangle AEFC inside it, which has a perimeter of 24.

AE = 8

BC = 5

What is the area of the parallelogram?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:17 Let's find the area of the parallelogram!

00:20 Remember, opposite sides are equal in a rectangle.

00:27 The perimeter is the total length of all sides added together.

00:36 Let's plug in the numbers and solve for E C.

00:55 We've found the length of E C. It's also the height of the parallelogram.

01:01 Now, we'll use the Pythagorean theorem in triangle E B C.

01:11 Let's substitute the numbers and find E B.

01:17 Next, we need to isolate E B.

01:24 Great! We now know the length of E B.

01:34 Let's calculate the area by multiplying the height by the base.

01:41 The side A B equals the sum of A E and E B.

01:45 Now we'll use the values to find the area of the parallelogram.

01:50 And there we have it! We've solved the problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The parallelogram ABCD contains the rectangle AEFC inside it, which has a perimeter of 24.

AE = 8

BC = 5

What is the area of the parallelogram?

Step-by-step solution

In the first step, we must find the length of EC, which we will identify with an X.

We know that the perimeter of a rectangle is the sum of all its sides (AE+EC+CF+FA),

Since in a rectangle the opposite sides are equal, the formula can also be written like this: 2AE=2EC.

We replace the known data:

$2\times8+2X=24$

$16+2X=24$

We isolate X:

$2X=8$

and divide by 2:

$X=4$

Now we can use the Pythagorean theorem to find EB.

(Pythagoras: $A^2+B^2=C^2$ )

$EB^2+4^2=5^2$

$EB^2+16=25$

We isolate the variable

$EB^2=9$

We take the square root of the equation.

$EB=3$

The area of a parallelogram is the height multiplied by the side to which the height descends, that is $AB\times EC$ .

$AB=\text{ AE}+EB$

$AB=8+3=11$

And therefore we will apply the area formula:

$11\times4=44$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rectangle Perimeter: Use formula 2(length + width) = 24 to find missing dimension
Pythagorean Theorem: Find height using $3^2 + 4^2 = 5^2$ in right triangle
Area Formula: Parallelogram area = base × height = 11 × 4 = 44 ✓

Common Mistakes

Avoid these frequent errors

Using the wrong base for area calculation
Don't use BC = 5 as the base for area calculation = wrong answer of 20! BC is a slanted side, not perpendicular to the height. Always use the horizontal base AB = 11 with perpendicular height EC = 4.

Practice Quiz

Test your knowledge with interactive questions

A parallelogram has a length equal to 6 cm and a height equal to 4.5 cm.

Calculate the area of the parallelogram.

FAQ

Everything you need to know about this question

Why can't I use BC = 5 as the base for the parallelogram area?

BC = 5 is a slanted side, not perpendicular to any height we can easily find. For area calculations, you need a base that's perpendicular to the height. Use AB = 11 as the base with height EC = 4.

How do I know that EC is the height of the parallelogram?

EC is perpendicular to both AB and the opposite side, making it the true height. In the rectangle AEFC, all angles are 90°, so EC forms right angles with the horizontal sides.

Why do I need the Pythagorean theorem in this problem?

The Pythagorean theorem helps find EB = 3 using the right triangle EBC where $EB^2 + EC^2 = BC^2$ . This gives us EB = 3, so AB = AE + EB = 8 + 3 = 11.

What if I calculated the rectangle's dimensions wrong?

Double-check using the perimeter formula: $2(8 + 4) = 24$ ✓. If AE = 8 and perimeter = 24, then EC must equal 4. Always verify your rectangle dimensions add up correctly!

Can I solve this without using the Pythagorean theorem?

No, you need the Pythagorean theorem to find EB = 3. Without it, you can't determine the full length AB = 11, which is essential for calculating the parallelogram's area.

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Calculate Parallelogram Area: Rectangle with Perimeter 24 Inside

Parallelogram Area with Rectangle Perimeter

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I use BC = 5 as the base for the parallelogram area?

How do I know that EC is the height of the parallelogram?

Why do I need the Pythagorean theorem in this problem?

What if I calculated the rectangle's dimensions wrong?

Can I solve this without using the Pythagorean theorem?

🌟 Unlock Your Math Potential

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