Pythagoras in 3D

The Pythagorean theorem is a fundamental concept in mathematics that relates the lengths of the sides of a right triangle. When applied in three dimensions, it allows us to find the distance or length within a three-dimensional space. Understanding how to calculate Pythagoras in 3D is essential in various fields, including physics, engineering, and computer graphics.

Prior Knowledge You need to be confident with Pythagoras' Theorem in 2D before working through this page.
⇩ Jump to Practice Questions ⇩

The space diagonal of a cuboid

\[ d = \sqrt{l^2 + w^2 + h^2} \]

Apply Pythagoras twice: across the base, then up to the opposite corner.

1. The cuboid (orientation)

A B C D E F G H l (length) h (height) w (width) AG EG

Cuboid ABCDEFGH. AG is the space diagonal we want; EG is the face diagonal we build first.

2. Base triangle: find the face diagonal

l = 6 cm w = 3 cm EG = ?

\(EG^2 = 6^2 + 3^2 = 45\)

3. Upright triangle: find the space diagonal

EG = √45 h = 2 cm AG = ?

\(AG^2 = 2^2 + 45 = 49,\ \ AG = 7\text{ cm}\)

What Is Pythagoras in 3D?

In a cuboid with length \(l\), width \(w\), and height \(h\), Pythagoras' Theorem is applied twice to find the length of the space diagonal \(d\).

Step 1 finds the face diagonal across the base: \(d_{\text{face}}^2 = l^2 + w^2\)

Step 2 uses that face diagonal as the base of a new right-angled triangle with the height:

\[ d^2 = l^2 + w^2 + h^2 \qquad \Rightarrow \qquad d = \sqrt{l^2 + w^2 + h^2} \]

This formula works for any cuboid or rectangular box. The key insight is that you never need to apply the theorem more than twice: always draw the two right-angled triangles separately.

Two Ways to Solve It

For the cuboid in Example 1 (length 6 cm, width 3 cm, height 2 cm, find AG) you can use either method. Both give the same answer.

Method A Two-Stage (Pythagoras Twice)

Extract two flat triangles from the cuboid (the same two shown in the stepped diagram above). Apply Pythagoras to each one separately, carrying the exact value between stages.

Stage 1: Base face
Triangle EGH on the base (Panel 2 above). Find EG.
\[EG^2 = 6^2 + 3^2 = 45\]
Keep \(EG^2 = 45\) exact. Do not square root yet.
Stage 2: Upright triangle
Triangle AEG standing upright (Panel 3 above). Find AG.
\[AG^2 = 2^2 + 45 = 49\]
\[AG = \sqrt{49} = 7\text{ cm}\]
Method B Formula Shortcut

Use the space diagonal formula directly. This combines both stages into one line, but you must still know how to derive it.

Space diagonal formula:
\[d = \sqrt{l^2 + w^2 + h^2}\]
Substitute \(l=6,\ w=3,\ h=2\):
\[AG = \sqrt{6^2 + 3^2 + 2^2} = \sqrt{36 + 9 + 4} = \sqrt{49}\]
\[AG = 7\text{ cm}\]
⚠️ When to use this: Only when finding the full space diagonal. For face diagonals, or when the question involves an intermediate length, you still need Method A.

Core Ideas

Two stages Find a face diagonal first, then use it as a side in a second triangle to find the space diagonal.
The formula For a cuboid: \(d = \sqrt{l^2 + w^2 + h^2}\). Memorise it, but also know how to derive it.
Draw separately Always redraw each right-angled triangle flat on paper. 3D problems become 2D problems this way.
Keep full precision Do not round between steps. Carry all decimal places until the final answer, then round to the required accuracy.

Worked Examples

💡 Space Diagonal of a Cuboid

Cuboid ABCDEFGH has length 6 cm, width 3 cm, and height 2 cm. Find the length of the space diagonal AG, giving your answer to 3 significant figures.

6 cm2 cm3 cmAG
1
Find the face diagonal EG across the base Draw triangle EGH on the base face.
\[EG^2 = 6^2 + 3^2 = 36 + 9 = 45\]
2
Use EG in triangle AEG to find AG Draw triangle AEG upright, with AE = 2 cm (height).
\[AG^2 = AE^2 + EG^2 = 2^2 + 45 = 4 + 45 = 49\]
3
Square root for the answer
\[AG = \sqrt{49} = 7 \text{ cm}\]

💡 Diagonal of a Cube

A cube has side length 8 cm. Find the length of its space diagonal to 3 significant figures.

8 cm8 cm8 cm?
1
All sides equal, so use the formula directly With \(l = w = h = 8\):
\[d^2 = 8^2 + 8^2 + 8^2 = 64 + 64 + 64 = 192\]
2
Square root for the answer
\[d = \sqrt{192} = 13.9 \text{ cm (3 s.f.)}\]

💡 Finding a Missing Dimension

A cuboid has length 12 cm, width 3 cm, and a space diagonal of 13 cm. Find the height of the cuboid.

12 cm?3 cm13 cm
1
Write the formula and substitute known values
\[d^2 = l^2 + w^2 + h^2\]
\[13^2 = 12^2 + 3^2 + h^2\]
2
Rearrange for \(h^2\)
\[h^2 = 169 - 144 - 9 = 16\]
3
Square root for the answer
\[h = \sqrt{16} = 4 \text{ cm}\]

💡 Real-Life Problem: Room Diagonal

A room is 12 m long, 4 m wide, and 3 m high. Find the distance from one corner of the floor to the opposite corner of the ceiling. Give your answer to 3 significant figures.

12 m3 m4 m?
1
Identify the dimensions \(l = 12\) m, \(w = 4\) m, \(h = 3\) m. The required length is the space diagonal.
2
Apply the formula
\[d^2 = 12^2 + 4^2 + 3^2 = 144 + 16 + 9 = 169\]
3
Square root for the answer
\[d = \sqrt{169} = 13 \text{ m}\]

🔑 Key Points

  • Apply Pythagoras twice: base face first, then use that diagonal with the height.
  • The space diagonal formula: \(d = \sqrt{l^2 + w^2 + h^2}\).
  • Always draw the two right-angled triangles separately.
  • Keep full calculator precision between steps; round only the final answer.

⚠️ Common Pitfalls

  • Rounding the face diagonal before using it in step 2; this compounds errors.
  • Trying to find the space diagonal in one step without drawing the two triangles.
  • Using the wrong edge as the height of the upright triangle.
  • Confusing the space diagonal with a face diagonal; they are different lengths.
⇩ Jump to Practice Questions ⇩ Next Topic → Using Trigonometry to Calculate Height

Pythagoras in 3D: Practice

Practise calculating Pythagoras in 3D in Room 1, with a 2D Pythagoras warm-up in Room 0. Type each length as a decimal rounded to 3 significant figures (for example 13.2); give only the final length, never a rounded intermediate. Each room runs across four tiers, left to right: Starter, Builder, Challenger, Master.

✓ Correct 0
✗ Re-attempts 0
🔥 Streak 0
🏆 Best 0

Difficulty rises left to right (Starter → Builder → Challenger → Master) and down each column. Answers are accepted to 3 significant figures with a small tolerance for rounding.

Video guide on how to calculate Pythagorus in 3D

This YouTube video provides a comprehensive explanation of calculating Pythagoras in 3D. The video includes visual representations and worked examples, allowing you to understand the process and applications of Pythagoras in three dimensions. Take the time to watch the video, and you’ll gain a solid understanding of calculating Pythagoras in 3D.