Find the Space Diagonal Formula: Expressing Length in Terms of x, y, and z

Q: How is this different from a rectangle's diagonal?

A rectangle's diagonal only needs $ \sqrt{length^2 + width^2} $ because it's 2D. For a rectangular prism (3D), you must add the third dimension: $ \sqrt{x^2 + y^2 + z^2} $.

Q: Do I need to simplify the square root?

Yes, when possible! If $ \sqrt{x^2 + y^2 + z^2} $ equals something like $ \sqrt{36} $, simplify to 6. Otherwise, leave it in radical form.

Space Diagonal Formula with Three Dimensions

Look at the rectangular prism in the figure.

Express the length of the diagonal in terms of x, y, and z.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's express the diagonal of the box using its dimensions.

00:11 Each face of the box is a rectangle, so all the angles are right angles.

00:17 Now, we'll use the Pythagorean Theorem in triangle E H G to find the length of E G.

00:24 This gives us the expression for the diagonal of face E G.

00:28 Remember, in rectangles, opposite sides are equal. So, each face in the box maintains this property.

00:38 There's a right angle here because a perpendicular to a face is perpendicular to any line going through it.

00:44 Next, use the Pythagorean Theorem in triangle E G C to find the length of E C.

00:53 Let's substitute the value of E G that we previously found.

01:01 Go ahead and take the square root.

01:08 And that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the rectangular prism in the figure.

Express the length of the diagonal in terms of x, y, and z.

Step-by-step solution

To solve for the diagonal of a rectangular prism with dimensions $x$ , $y$ , and $z$ , we'll utilize the Pythagorean theorem in three dimensions. This enables us to account for the three different sides of the prism.

Let us break this down into steps:

Step 1: Identify the dimensions given: length $x$ , width $y$ , and height $z$ .
Step 2: Recognize that the diagonal stretches across the prism from one corner to the opposite corner, forming a 3D hypotenuse.
Step 3: Apply the formula for the diagonal of a rectangular prism:

\text{Diagonal} = \sqrt{x^2 + y^2 + z^2}

This formula arises because the diagonal spans across the 3D space of the prism. By applying the Pythagorean theorem first to the base rectangle and then incorporating the height, we account for all dimensions of the prism.

Thus, the length of the diagonal is given by $\sqrt{x^2 + y^2 + z^2}$ .

Final Answer

$\sqrt{x^2+y^2+z^2}$

Key Points to Remember

Essential concepts to master this topic

3D Pythagorean Theorem: Space diagonal uses all three dimensions: length, width, height
Formula Application: $d = \sqrt{x^2 + y^2 + z^2}$ for dimensions x, y, z
Verification Check: Each dimension squared appears once in the formula under the radical ✓

Common Mistakes

Avoid these frequent errors

Using only two dimensions in the formula
Don't use $\sqrt{x^2 + y^2}$ for a 3D diagonal = flat diagonal only! This ignores the third dimension and gives a shorter length than the actual space diagonal. Always include all three dimensions: $\sqrt{x^2 + y^2 + z^2}$ .

Practice Quiz

Test your knowledge with interactive questions

Consider a right-angled triangle, AB = 8 cm and AC = 6 cm.
Calculate the length of side BC.

FAQ

Everything you need to know about this question

Why can't I just add x + y + z for the diagonal?

The diagonal doesn't follow the edges of the prism - it cuts straight through the interior! Adding dimensions gives you the perimeter path, not the direct distance. You need the Pythagorean theorem for the shortest path.

How is this different from a rectangle's diagonal?

A rectangle's diagonal only needs $\sqrt{length^2 + width^2}$ because it's 2D. For a rectangular prism (3D), you must add the third dimension: $\sqrt{x^2 + y^2 + z^2}$ .

What if two dimensions are the same?

The formula still works perfectly! If x = y = 5 and z = 3, then diagonal = $\sqrt{5^2 + 5^2 + 3^2} = \sqrt{59}$ . Each dimension gets squared separately.

Do I need to simplify the square root?

Yes, when possible! If $\sqrt{x^2 + y^2 + z^2}$ equals something like $\sqrt{36}$ , simplify to 6. Otherwise, leave it in radical form.

Can I use this formula for a cube?

Absolutely! For a cube with side length s, the space diagonal is $\sqrt{s^2 + s^2 + s^2} = \sqrt{3s^2} = s\sqrt{3}$ . It's a special case of the general formula.

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