Diagonal of Cube with Face and Space Diagonal Formulas

Diagonal of Cube refers to the line joining two opposite vertices through the interior of a cube. For a cube with side length a, its space diagonal is a√3. This page covers the derivation, face diagonal relation, volume-based calculations, solved examples and common aptitude problems involving cube diagonals.

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What Is the Diagonal of a Cube?

The diagonal of a cube is the line segment joining two opposite vertices that do not lie on the same face. If the side of the cube is a, its space diagonal is a√3.

A cube has equal edges and three mutually perpendicular dimensions. Applying the three-dimensional Pythagorean theorem gives d² = a² + a² + a² = 3a², so d = a√3. The diagonal is measured in the same unit as the side.

Diagonal of Cube Video Lessons

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Cube Volume, Surface Area and Diagonal

Learn how to calculate a cube’s volume, surface area, and diagonal using the relevant formulas and understand the relationship between its dimensions and measurements.

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Diagonal of Cube Formula & Tricks

Important Formulas

Space diagonal
d = a√3

Here, a is the side length and d is the diagonal joining opposite vertices through the cube.

Square of the space diagonal
d² = 3a²

This form is useful when the question asks for the square of the diagonal or provides squared lengths.

Face diagonal
f = a√2

The diagonal of any square face is found using the two-dimensional Pythagorean theorem.

Diagonal from volume
d = √3 ∛V

Since the volume of a cube is V = a³, its side is ∛V and its space diagonal is √3∛V.

Side from diagonal
a = d/√3 = d√3/3

Rationalising the denominator gives the equivalent form d√3/3.

Quick Tricks

Use the √3 multiplier directly

For a cube, multiply the side by √3. There is no need to calculate the three-dimensional path separately.

Example: If a = 6 cm, then d = 6√3 cm.
Convert a volume into a side first

Take the cube root of the volume, then multiply the result by √3.

Example: For V = 125 cm³, a = ∛125 = 5 cm, so d = 5√3 cm.
Use the squared relation for comparisons

When comparing diagonals of cubes, compare d² = 3a². Thus, the ratio of diagonals equals the ratio of their sides.

Example: Cubes with sides 4 cm and 10 cm have diagonals in the ratio 4:10 = 2:5.

Diagonal of Cube Concepts

Derivation Using the Face Diagonal

The space diagonal is obtained by applying the Pythagorean theorem first on a face and then in the third dimension.

The face diagonal is f = √(a² + a²) = a√2. The space diagonal then satisfies d² = f² + a² = 2a² + a² = 3a², giving d = a√3.

Example: For a side of 8 cm, the face diagonal is 8√2 cm and the space diagonal is 8√3 cm.

Difference Between Face and Space Diagonal

A face diagonal joins opposite vertices of one square face, whereas a space diagonal joins opposite vertices through the interior of the cube.

For side a, the face diagonal is a√2 and the space diagonal is a√3. Therefore, the space diagonal is longer by a factor of √(3/2) compared with the face diagonal.

Example: If the face diagonal is 10√2 cm, then a = 10 cm and the space diagonal is 10√3 cm.

Finding the Diagonal from Volume

When the volume is given, find the side by taking its cube root and then use d = a√3.

For volume V, a³ = V, so a = ∛V. Substitution gives d = √3∛V. The volume must be expressed in cubic units before taking the cube root.

Example: If V = 216 m³, then a = ∛216 = 6 m and d = 6√3 m.

Finding Side or Volume from the Diagonal

If the space diagonal is known, the side is d/√3 and the volume is (d/√3)³.

Using d = a√3, we get a = d/√3. Hence, V = a³ = d³/(3√3), which may also be written as d³√3/9 after rationalisation.

Example: If d = 9√3 cm, then a = 9 cm and V = 9³ = 729 cm³.

Diagonal of Cube Visual Guide & Data

Use the table and chart below to understand the data better.

Cube Diagonal Formula Comparison

Formulas for the face diagonal and space diagonal of a cube with side a.

Measurement Formula Geometric location
Face diagonal a√2 Joins opposite vertices of one square face
Space diagonal a√3 Joins opposite vertices through the cube
Side from space diagonal d/√3 Reverses the space-diagonal formula

Note: The space diagonal is longer than the face diagonal for the same cube.

Space Diagonal of a Cube

Visual guide: Space Diagonal of a Cube
Side: a Space diagonal: d = a√3
The space diagonal joins two opposite vertices of a cube and has length d = a√3.

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Practice Diagonal of Cube Questions

Practise published questions related to this topic.

1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Diagonal of Cube Revision Points

Key formulas and rules for quick revision.

  • Space diagonal of a cube: d = a√3.
  • Face diagonal of a cube: f = a√2.
  • The derivation is d² = a² + a² + a² = 3a².
  • From volume V, the diagonal is d = √3∛V.
  • From diagonal d, the side is a = d/√3.
  • The diagonal and side have the same unit; volume must be in cubic units.

Related Topic Explainers

Area of Square → Perimeter of Square → Area of Rectangle → Perimeter of Rectangle → Area of Triangle → Perimeter of Triangle → Heron's Formula → Area of Triangle Using Two Sides and Included Angle →

Diagonal of Cube FAQs

What is the diagonal of a cube with side 7 cm?

Using d = a√3, d = 7√3 cm.

What is the diagonal of a cube with volume 64 cm³?

The side is ∛64 = 4 cm. Therefore, the space diagonal is 4√3 cm.

What is the difference between a cube's face diagonal and space diagonal?

The face diagonal is a√2 and lies on one square face. The space diagonal is a√3 and passes through the cube's interior between opposite vertices.

If the space diagonal of a cube is 12√3 cm, what is its side?

a = d/√3 = 12√3/√3 = 12 cm.

What is the space diagonal when the face diagonal is 6√2 cm?

Since the face diagonal is a√2, the side is 6 cm. The space diagonal is therefore 6√3 cm.

What is the volume of a cube whose space diagonal is 6√3 cm?

The side is a = 6√3/√3 = 6 cm. Hence, the volume is 6³ = 216 cm³.

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