Calculate Rectangle Side Length: Given Area 24 cm² and Partial Measurements

Pythagorean Theorem with Rectangle Area Calculations

Shown below is the rectangle ABCD.

Given in cm:

AK = 5

DK = 4

The area of the rectangle is 24 cm².

Calculate the side AB.

S=24S=24S=24555444AAABBBCCCDDDKKK

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:14 To start, we need to calculate the length of side A B.
00:19 First, we'll use the Pythagorean Theorem in triangle A D K to find side A D.
00:28 Let's substitute the given values to solve for A D.
00:46 Now, we need to isolate A D to find its length.
00:58 Great! We have found the length of side A D.
01:06 Next, let's calculate the area of the rectangle using the formula: side multiplied by side.
01:15 Substitute the known values to find side A B.
01:28 Now, let's isolate A B to determine its length.
01:38 And there you go! We've solved the problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Shown below is the rectangle ABCD.

Given in cm:

AK = 5

DK = 4

The area of the rectangle is 24 cm².

Calculate the side AB.

S=24S=24S=24555444AAABBBCCCDDDKKK

2

Step-by-step solution

Let's look at triangle ADK in order to calculate side AD:

AD2+DK2=AK2 AD^2+DK^2=AK^2

Now let's substitute in our values:

AD2+42=52 AD^2+4^2=5^2

AD2+16=25 AD^2+16=25

We'll then move 16 to the other side and change the sign to the appropriate one:

AD2=2516 AD^2=25-16

AD2=9 AD^2=9

Next, we'll take the square root and get:

AD=3 AD=3

Since AD is a side of rectangle ABCD, we can calculate side AB as follows:

S=AB×AD S=AB\times AD

Let's substitute in our values:

24=3×AB 24=3\times AB

Finally, we'll divide both sides by 3:

AB=8 AB=8

3

Final Answer

8

Key Points to Remember

Essential concepts to master this topic
  • Pythagorean Theorem: Use a2+b2=c2 a^2 + b^2 = c^2 to find missing rectangle side
  • Technique: Calculate AD2+42=52 AD^2 + 4^2 = 5^2 , so AD=3 AD = 3 cm
  • Check: Verify area: AB×AD=8×3=24 AB \times AD = 8 \times 3 = 24 cm² ✓

Common Mistakes

Avoid these frequent errors
  • Using rectangle area formula before finding all side lengths
    Don't substitute unknown values directly into Area = length × width = 24! This creates equations with two unknowns that are harder to solve. Always use the Pythagorean theorem first to find the missing side length, then use the 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?
666444AAABBBCCCDDD

FAQ

Everything you need to know about this question

Why can't I just use the area formula directly to find AB?

+

Because you need both dimensions of the rectangle to use Area = length × width. You only know the area (24 cm²) but need to find AD first using the Pythagorean theorem with triangle ADK.

How do I know which triangle to use for the Pythagorean theorem?

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Look for a right triangle where you know two sides and need the third. Here, triangle ADK is a right triangle with AK = 5 (hypotenuse) and DK = 4, so you can find AD.

What if I get a negative answer when taking the square root?

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In geometry problems, lengths are always positive. When you calculate 9=3 \sqrt{9} = 3 , only use the positive value since we're measuring a physical distance.

How can I be sure K is at a right angle?

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The problem shows K on side DC of the rectangle, and diagonal AK creates a right triangle with the rectangle's sides. Rectangle corners are always 90°, making ADK a right triangle.

Can I solve this problem a different way?

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Yes! You could also use coordinate geometry by placing the rectangle on a coordinate plane, but the Pythagorean theorem approach is more direct and requires fewer calculations.

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