CDS Mathematics · Mensuration 3D

Diagonals & Cuboid Identities

The space diagonal of a cuboid, the longest rod in a room, and the algebraic identities that link the sum of the edges, the diagonal, the surface area and the volume.

All Mensuration 3D notes CDS Mathematics strategy

Why this matters

These questions look as if they need the three edges, and almost never do. One identity — the square of the sum of the edges — turns a diagonal and a surface area into the sum of the edges and back again.

Concept 1 of 3: The space diagonal

The longest straight line inside a cuboid runs from one corner to the opposite one. It is Pythagoras used twice: first across the floor, then up.

Definition

Space diagonal $= \sqrt{l^2 + b^2 + h^2}$ ; floor diagonal $= \sqrt{l^2 + b^2}$ .
Cube of edge $a$ : face diagonal $a\sqrt2$ , space diagonal $a\sqrt3$ , so surface $= 6a^2 = 2d^2$ .
Longest rod in a room $=$ the space diagonal; longest rod on the floor $=$ the floor diagonal. Their squares differ by $h^2$ .
The angle between the space diagonal and the floor diagonal: $\cos\alpha = \dfrac{\text{floor diagonal}}{\text{space diagonal}}$ .

Space diagonal

d = \sqrt{l^2 + b^2 + h^2}

Worked example

The longest rod that fits in a room is

13

m and the longest rod that lies on its floor is

12

m. Find the room's height.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (II) 2021 — Elementary Mathematics · Q66Easy

Example 1 · Mensuration 3D · Diagonals and Cuboid Identities

The length and breadth of a room are 21 m and 16 m respectively. If the length of the longest rod that can be placed in the room is 29 m, then what is the height of the room ?

The largest slice is not a face

A plane through two opposite edges of a cube cuts a rectangle

a\times a\sqrt2

, larger than a face (

a^2

) and larger than the hexagonal section through the centre.

Drill 8 more on the space diagonal

Concept 2 of 3: Sum of edges, surface area and diagonal

Squaring

l + b + h

gives the diagonal squared plus the surface area. Any two of the three quantities give the third, and the edges themselves are never needed.

Definition

$(l + b + h)^2 = (l^2 + b^2 + h^2) + 2(lb + bh + hl)$ , i.e. $\text{sum}^2 = d^2 + \text{surface}$ .
The cube identity: $l^3 + b^3 + h^3 - 3lbh = (l + b + h)\left[(l^2 + b^2 + h^2) - (lb + bh + hl)\right]$ .

The square of the sum

(l + b + h)^2 = d^2 + S

Worked example

A cuboid's edges add to

19

cm and its diagonal is

13

cm. Find its total surface area.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2023 · CDS (I) 2023 — Elementary Mathematics · Q60Moderate

Example 2 · Mensuration 3D · Diagonals and Cuboid Identities

The length of a diagonal of a cuboid is 11 cm. The surface area is 240 square cm. What is the sum of its length, breadth and height ?

Drill 2 more on sum of edges, surface area and diagonal

Concept 3 of 3: Face areas, volume and reciprocals

The three faces meeting at a corner have areas

lb

bh

and

hl

. Multiply them and every edge appears twice: the product is the square of the volume.

Definition

Adjacent faces $x, y, z$ : $xyz = (lbh)^2 = V^2$ .
$\dfrac1l + \dfrac1b + \dfrac1h = \dfrac{lb + bh + hl}{lbh} = \dfrac{S}{2V}$ .

Adjacent faces

xyz = V^2, \qquad \frac1l + \frac1b + \frac1h = \frac{S}{2V}

Worked example

Three faces of a cuboid meeting at a corner have areas

12

15

and

20

^2

. Find its volume.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2023 · CDS (I) 2023 — Elementary Mathematics · Q87Moderate

Example 3 · Mensuration 3D · Diagonals and Cuboid Identities

The volume of a cuboid is 3600 cubic cm. The areas of two adjacent faces are 225 square cm and 144 square cm. What is the area of the other adjacent face ?

Drill 3 more on face areas, volume and reciprocals

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

The space diagonal
Space diagonal
$d = \sqrt{l^2 + b^2 + h^2}$
Sum of edges, surface area and diagonal
The square of the sum
$(l + b + h)^2 = d^2 + S$
Face areas, volume and reciprocals
Adjacent faces
$xyz = V^2, \qquad \frac1l + \frac1b + \frac1h = \frac{S}{2V}$

Watch out for (1)

The largest slice is not a face
→ The space diagonal

Test yourself on Mensuration 3D

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Mensuration 3D chapter test Drill all 16 questions