Calculate Rectangle Perimeter: Finding the Distance Around a 6 by 10 Shape

Q: Why can't I just add all three measurements: 6 + 8 + 10?

The diagonal is not part of the perimeter! Perimeter measures the distance around the outside of a shape. The diagonal cuts through the middle and doesn't form part of the boundary.

Q: What if I don't remember the Pythagorean theorem?

For a right triangle with sides a, b and diagonal c: $ a^2 + b^2 = c^2 $. Think of it as: "square the two sides, add them, then take the square root to find the third side."

Q: Why is the answer 28 and not 32?

32 comes from incorrectly adding the diagonal: 6 + 8 + 10 + 8 = 32. But perimeter only uses the four outer sides: 8 + 6 + 8 + 6 = 28, or more simply $ 2(8 + 6) = 28 $.

Q: Can I solve this without finding the missing side first?

No! You must find the missing side length before calculating perimeter. The diagram only shows the height (6) and diagonal (10), so you need the Pythagorean theorem to find the base (8).

Rectangle Perimeter with Pythagorean Calculations

Calculate the perimeter of the rectangle shown below:

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the perimeter of the rectangle

00:03 Let's use the Pythagorean theorem in triangle BCD

00:17 Substitute appropriate values according to the given data and solve for DC

00:30 Isolate DC

00:45 This is the value of side DC

00:50 Opposite sides are equal in a rectangle

01:04 The perimeter of the rectangle equals the sum of its sides

01:13 Substitute appropriate values and solve for the perimeter

01:37 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

Calculate the perimeter of the rectangle shown below:

Step-by-step solution

Let's solve for the perimeter of the rectangle by following these steps:

First, identify the given information: The diagonal $AC$ is 10 units and the height $BC$ is 6 units.
Next, use the Pythagorean theorem to calculate the missing side (the base), $AB$ . Given that $AB^2 + BC^2 = AC^2$ , we can write:
Step 1: Substitute known values into the equation: $AB^2 + 6^2 = 10^2$ .
Step 2: Calculate $AB^2$ :

AB^2 + 36 = 100\\ AB^2 = 100 - 36\\ AB^2 = 64\\ AB = \sqrt{64} = 8

We have found the base $AB = 8$ units.

Now, calculate the perimeter $P$ using the perimeter formula $P = 2(l + w)$ , where $l$ is the length (AB = 8) and $w$ is the width (BC = 6).

P = 2(8 + 6) = 2 \times 14 = 28

Therefore, the perimeter of the rectangle is 28 units.

The correct choice is 3: 28.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: Rectangle perimeter = 2(length + width) for all sides
Technique: Use Pythagorean theorem: $AB^2 + 6^2 = 10^2$ gives AB = 8
Check: Verify diagonal: $\sqrt{8^2 + 6^2} = \sqrt{100} = 10$ ✓

Common Mistakes

Avoid these frequent errors

Using diagonal as a side in perimeter formula
Don't add the diagonal (10) to the perimeter calculation = wrong answer 32! The diagonal is inside the rectangle, not part of the boundary. Always use only the four outer sides: 2(8 + 6) = 28.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle below.

Side DC has a length of 1.5 cm and side AD has a length of 9.5 cm.

What is the perimeter of the rectangle?

FAQ

Everything you need to know about this question

Why can't I just add all three measurements: 6 + 8 + 10?

The diagonal is not part of the perimeter! Perimeter measures the distance around the outside of a shape. The diagonal cuts through the middle and doesn't form part of the boundary.

How do I know which sides are length and width?

It doesn't matter which you call length or width! In our rectangle, you can use either $2(8 + 6)$ or $2(6 + 8)$ - both give the same answer: 28.

What if I don't remember the Pythagorean theorem?

For a right triangle with sides a, b and diagonal c: $a^2 + b^2 = c^2$ . Think of it as: "square the two sides, add them, then take the square root to find the third side."

Why is the answer 28 and not 32?

32 comes from incorrectly adding the diagonal: 6 + 8 + 10 + 8 = 32. But perimeter only uses the four outer sides: 8 + 6 + 8 + 6 = 28, or more simply $2(8 + 6) = 28$ .

Can I solve this without finding the missing side first?

No! You must find the missing side length before calculating perimeter. The diagram only shows the height (6) and diagonal (10), so you need the Pythagorean theorem to find the base (8).

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