Calculate the Diagonal of a Rectangular Prism: Dimensions a/2, 3b, and a+b

Q: Why do we square each dimension first?

The 3D diagonal formula comes from applying the Pythagorean theorem twice - once for the base diagonal, then again for the space diagonal. Each application requires squaring the sides!

Q: How do I expand (a+b)² correctly?

Use the pattern $ (a+b)^2 = a^2 + 2ab + b^2 $. Don't forget the middle term 2ab - it's the most commonly missed part!

Q: What do I do with the fraction a/2?

Square it like any other term: $ \left(\frac{a}{2}\right)^2 = \frac{a^2}{4} $. Remember that squaring a fraction means squaring both numerator and denominator.

Q: How do I combine a² and a²/4?

Convert to the same denominator: $ a^2 = \frac{4a^2}{4} $, so $ a^2 + \frac{a^2}{4} = \frac{4a^2 + a^2}{4} = \frac{5a^2}{4} $

Q: Can I simplify the final answer further?

The answer $ \sqrt{\frac{5a^2}{4} + 2ab + 10b^2} $ is already in simplest form. You cannot factor out terms from under a square root unless they form perfect squares.

3D Diagonal Formula with Algebraic Expressions

Look at the rectangular prism below.

Its height is $\frac{a}{2}$ , its length is $3b$ , and its width is $a+b$ .

Calculate the diagonal of the rectangular prism.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:12 Let's calculate the diagonal of the box.

00:16 First, draw the face diagonal and label it as X.

00:21 Now, we'll use the Pythagorean theorem to find the expression fo r this diagonal X.

00:37 Next, let's collect all the terms and rearrange them.

00:42 Great! This gives us the expression for the face diagonal.

00:50 Draw a triangle where the hypotenuse is the box's diagonal and label it as Y.

00:58 Again, use the Pythagorean theorem to find the expression fo r the box diagonal Y.

01:05 Substitute the value of X that we found earlier into this expression .

01:16 Rearrange and simplify the terms once more.

01:21 And that's how we find the solution to this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the rectangular prism below.

Its height is $\frac{a}{2}$ , its length is $3b$ , and its width is $a+b$ .

Calculate the diagonal of the rectangular prism.

Step-by-step solution

To calculate the diagonal of the rectangular prism, we use the formula for the diagonal $d$ in a cuboid: $d = \sqrt{l^2 + w^2 + h^2}$ where $l = 3b$ , $w = a+b$ , and $h = \frac{a}{2}$ .

First, we compute each squared term:

Length squared: $(3b)^2 = 9b^2$
Width squared: $(a+b)^2 = a^2 + 2ab + b^2$
Height squared: $\left(\frac{a}{2}\right)^2 = \frac{a^2}{4}$

Now, adding these values: $l^2 + w^2 + h^2 = 9b^2 + (a^2 + 2ab + b^2) + \frac{a^2}{4}$ Combine the terms: $= 9b^2 + a^2 + 2ab + b^2 + \frac{a^2}{4}$ Simplify: $= \frac{a^2}{4} + a^2 + 10b^2 + 2ab$ Notice that $a^2 + \frac{a^2}{4} = \frac{4a^2}{4} + \frac{a^2}{4} = \frac{5a^2}{4}$ .

Thus, the diagonal is: $d = \sqrt{\frac{5a^2}{4} + 2ab + 10b^2}$ This expression matches choice 1 in the given multiple-choice answers.

Therefore, the solution to the problem is $\sqrt{\frac{5}{4}a^2 + 2ab + 10b^2}$ .

Final Answer

$\sqrt{\frac{5}{4}a^2+2ab+10b^2}$

Key Points to Remember

Essential concepts to master this topic

Formula: 3D diagonal uses $d = \sqrt{l^2 + w^2 + h^2}$
Technique: Expand $(a+b)^2 = a^2 + 2ab + b^2$ carefully
Check: Combine like terms: $a^2 + \frac{a^2}{4} = \frac{5a^2}{4}$ ✓

Common Mistakes

Avoid these frequent errors

Adding dimensions instead of their squares
Don't add a/2 + 3b + (a+b) = diagonal! This gives a linear expression, not the diagonal length. The diagonal formula requires squaring each dimension first, then adding, then taking the square root. Always use $d = \sqrt{l^2 + w^2 + h^2}$ .

Practice Quiz

Test your knowledge with interactive questions

Look at the triangle in the diagram. How long is side AB?

FAQ

Everything you need to know about this question

Why do we square each dimension first?

The 3D diagonal formula comes from applying the Pythagorean theorem twice - once for the base diagonal, then again for the space diagonal. Each application requires squaring the sides!

How do I expand (a+b)² correctly?

Use the pattern $(a+b)^2 = a^2 + 2ab + b^2$ . Don't forget the middle term 2ab - it's the most commonly missed part!

What do I do with the fraction a/2?

Square it like any other term: $\left(\frac{a}{2}\right)^2 = \frac{a^2}{4}$ . Remember that squaring a fraction means squaring both numerator and denominator.

How do I combine a² and a²/4?

Convert to the same denominator: $a^2 = \frac{4a^2}{4}$ , so $a^2 + \frac{a^2}{4} = \frac{4a^2 + a^2}{4} = \frac{5a^2}{4}$

Can I simplify the final answer further?

The answer $\sqrt{\frac{5a^2}{4} + 2ab + 10b^2}$ is already in simplest form. You cannot factor out terms from under a square root unless they form perfect squares.

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