Calculate Rectangle Perimeter: Area Ratio 1/3 Between Triangle and Rectangle

Q: How does the triangle area help find rectangle dimensions?

Once you know triangle area = 24, and it's $ \frac{1}{3} $ of rectangle area, multiply: rectangle area = 3 × 24 = 72. Then use Area = length × width to find the unknown side.

Q: How do I calculate the final perimeter?

In a rectangle, opposite sides are equal. So AB = CD = 12 and AD = BC = 6. Perimeter = $ 12 + 6 + 12 + 6 = 36 $.

Q: What if I got a different triangle area calculation?

Double-check your Pythagorean theorem: $ EC^2 = 10^2 - 6^2 = 100 - 36 = 64 $, so EC = 8. Then triangle area = $ \frac{1}{2} \times 6 \times 8 = 24 $.

Rectangle Perimeter with Area Ratios

Observe the following rectangle:

The the area of the triangle ΔBCE is $\frac{1}{3}$ the area of the rectangle ABCD.

Calculate the perimeter of the rectangle ABCD.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's calculate the perimeter of rectangle ABCD.

00:14 Now, apply the Pythagorean theorem to triangle B C E.

00:18 Next, substitute the values into the equation and solve for E C.

00:36 Alright, let's isolate E C.

00:47 This gives us the length of side E C.

00:52 Now, use the triangle area formula, which is height times base divided by 2.

00:58 Plug in the values and solve for the triangle's area.

01:03 This result is the area of the triangle.

01:14 Remember, the triangle's area equals one-third of the rectangle's area.

01:19 Time to isolate the rectangle's area.

01:30 Now, we have the rectangle's area.

01:43 Use the rectangle area formula: length times width.

01:52 Insert values here, then find D C.

01:56 Let's isolate the side D C.

02:01 This is the length of side D C.

02:08 Remember, opposite sides in a rectangle are equal.

02:13 Now, calculate the perimeter: sum of all sides.

02:24 Substitute values and solve for the perimeter.

02:33 And there you have it! This is how we solve for the perimeter.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Observe the following rectangle:

The the area of the triangle ΔBCE is $\frac{1}{3}$ the area of the rectangle ABCD.

Calculate the perimeter of the rectangle ABCD.

Step-by-step solution

Observe triangle BCE and proceed to calculate side EC using the Pythagorean theorem:

$BC^2+EC^2=BE^2$

Insert the known values into the theorem:

$6^2+EC^2=10^2$

$36+EC^2=100$

$EC^2=100-36$

$EC^2=64$

Determine the square root:

$EC=8$

Calculate the area of triangle BCE:

$S=\frac{BC\times EC}{2}$

Insert the known values once again:

$S=\frac{6\times8}{2}=\frac{48}{2}=24$

According to the given data, the area of triangle BCE is one-third of rectangle ABCD's area, therefore:

$24=\frac{1}{3}$

Multiply by 3:

$S=3\times24=72$

The area of the rectangle equals 72

Now let's determine side CD

We know that the area of a rectangle equals the length multiplied by the width, meaning:

$S=BC\times DC$

Insert the known values in the formula:

$72=6\times CD$

Divide both sides by 6:

$CD=12$

Given that in a rectangle opposite sides are equal, AB also equals 12

Proceed to calculate the perimeter of the rectangle ABCD:

$12+6+12+6=24+12=36$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Area Ratio: Triangle area = $\frac{1}{3}$ rectangle area gives total rectangle area
Pythagorean Theorem: Find missing triangle side: $6^2 + EC^2 = 10^2$ gives EC = 8
Verification: Check perimeter: 2(12) + 2(6) = 30 matches calculated rectangle dimensions ✓

Common Mistakes

Avoid these frequent errors

Using triangle area as rectangle area
Don't assume triangle BCE area = 24 is the rectangle area! This gives wrong dimensions since 24 ÷ 6 = 4, making perimeter = 20. Always multiply triangle area by 3 first: rectangle area = 3 × 24 = 72.

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 do I need the Pythagorean theorem in a perimeter problem?

The diagram shows triangle BCE with diagonal BE = 10 and height BC = 6. To find the triangle's area, you need its base EC. Use Pythagorean theorem: $6^2 + EC^2 = 10^2$ to get EC = 8.

How does the triangle area help find rectangle dimensions?

Once you know triangle area = 24, and it's $\frac{1}{3}$ of rectangle area, multiply: rectangle area = 3 × 24 = 72. Then use Area = length × width to find the unknown side.

Which sides of the rectangle do I need to find?

You already know BC = 6 from the diagram. Use rectangle area = 72 and the formula Area = BC × CD to find: $72 = 6 \times CD$ , so CD = 12.

How do I calculate the final perimeter?

In a rectangle, opposite sides are equal. So AB = CD = 12 and AD = BC = 6. Perimeter = $12 + 6 + 12 + 6 = 36$ .

What if I got a different triangle area calculation?

Double-check your Pythagorean theorem: $EC^2 = 10^2 - 6^2 = 100 - 36 = 64$ , so EC = 8. Then triangle area = $\frac{1}{2} \times 6 \times 8 = 24$ .

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