How do you compute the diagonal of a cuboid?

With D = √(a² + b² + H²). It comes from applying the Pythagorean theorem twice: first the base diagonal d² = a² + b², then the space diagonal in the triangle with legs d and H, so D² = d² + H² = a² + b² + H². For a cube, where all edges are equal, this gives D = a√3.

Lines that have no point in common and are not parallel either — no single plane contains both. On a plane that case does not exist, which is why it is the first thing to accept in solid geometry. An example is an edge of the base of a cuboid together with a non-adjacent edge of its top face.

How is the apothem of a pyramid different from its lateral edge?

The apothem is the height of a lateral face and stands over half a base edge, so m² = H² + (a/2)². The lateral edge runs to a base vertex and stands over half the base diagonal, so b² = H² + (a√2 / 2)². They are two different right triangles sharing the height and differing in the other leg — confusing them is the commonest mistake in pyramid problems.

What is the area of the diagonal section of a cube?

The section through two opposite vertical edges is a rectangle with sides a and a√2, so its area is a²√2. For a 5 cm edge that is 25√2 ≈ 35.36 cm². The other standard section — through the three vertices adjacent to one corner — is an equilateral triangle of side a√2 and area a²√3 / 2.

Advanced level

Solid geometry: angles, segments and sections

Q: How is the angle between a line and a plane measured?

It is the angle between the line and its orthogonal projection onto the plane. In practice you find the right triangle whose hypotenuse is the segment in question, one leg its projection and the other the height. The angle then comes from the tangent: the height divided by the length of the projection.

The volume and the surface area of a solid are already settled. This lesson measures what is inside: the diagonal of a cuboid, the angle it makes with the base, the apothem and the lateral edge of a pyramid, and the sections of a cube. Every one of these is a single right triangle found in space.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Where this is used

Real situations where you count exactly the way this lesson teaches:

A box and the longest thing that fits in it
A carton with inside dimensions 60 × 40 × 30 cm takes an object longer than any of its edges: along the diagonal it fits √(60² + 40² + 30²) = √6100 ≈ 78 cm. That is the only number to compare with the length of a fishing rod, an easel or a length of trim before packing.
Roofing a pyramid roof
A pyramid roof over an 8 × 8 m square is 3 m high. The apothem — the height of one triangular plane — is √(3² + 4²) = 5 m, so one plane covers 8 · 5 / 2 = 20 m² and the whole roof 80 m², which is the figure the covering is ordered by.
Convert it here: Area
A mast and its guy wire
A guy wire fixed at the top of a 12 m mast and anchored 5 m from its foot is √(12² + 5²) = 13 m long. The angle it makes with the ground has tangent 12 / 5 = 2.4 — and that decides whether the anchor holds, because the closer to the mast, the larger the vertical pull.
A lift and carrying a board
A lift car measures 100 × 130 × 220 cm. A 250 cm board will not go in flat or upright, but the diagonal of the car is √(100² + 130² + 220²) ≈ 275 cm, so it goes in at a slant. The same calculation works for a stairwell and for a van’s load space.
Cutting a cubic block
A block of material with a 40 cm edge, cut along a diagonal of its base, gives a rectangle 40 by 40√2 ≈ 56.6 cm, a cut area of 40² · √2 ≈ 2263 cm². At a price per square centimetre of cut it is that number, not the area of a face, that lands on the invoice.

All formulas

Diagonal of a cuboid
$D^2 = a^2 + b^2 + H^2$
a and b are the base edges, H the height
Diagonal of a cube
$D = a\sqrt{3}$
the case a = b = c
Angle between the diagonal and the base
$\tan \alpha = \frac{H}{d}$
d is the diagonal of the base
Apothem of a pyramid
$m^2 = H^2 + \left(\frac{a}{2}\right)^2$
it stands over half a base edge
Lateral edge of a pyramid
$b^2 = H^2 + \left(\frac{a\sqrt{2}}{2}\right)^2$
it stands over half the base diagonal
Diagonal section of a cube
$P = a^2\sqrt{2}$
a rectangle with sides a and a√2

Volume and surface area say how much a solid holds and how much material it takes. This lesson is about something else: the segments and angles inside a solid — diagonals, apothems, edges and cutting planes.

The method is the same every time, and worth naming up front: find a right triangle in space and go back to the Pythagorean theorem. The whole difficulty of solid geometry is seeing where that triangle lies.

Lines in space

On a plane two lines either meet or are parallel. In space there is a third possibility:

Position	Common point	Common plane
intersecting	one	yes
parallel	none	yes
skew	none	no

Skew lines are what cannot be drawn on a sheet of paper: an edge of the bottom of a box and a non-adjacent edge of its lid never meet, and are not parallel either.

A line is perpendicular to a plane when it is perpendicular to two intersecting lines of that plane — and then it is perpendicular to every line in it. The height of a right pyramid and the lateral edge of a right prism are exactly like that.

The angle between a line and a plane is the angle between the line and its orthogonal projection onto the plane. That definition is what turns into a triangle in an exercise: the hypotenuse is the segment in question, one leg its projection, the other the height.

The diagonal of a cuboid

The base diagonal d and the space diagonal D. The triangle with legs d and H lies in the plane of the diagonal section.

The computation takes two steps. First the diagonal of the base:

d^2 = a^2 + b^2

Then the diagonal of the solid, in the triangle with legs $d$ and $H$ :

D^2 = d^2 + H^2 = a^2 + b^2 + H^2

Tables often write the same formula as $D^2 = a^2 + b^2 + c^2$ , with $c$ as the third edge — the same number under a different letter.

For a cube all edges are equal, so $D^2 = 3a^2$ , that is

D = a\sqrt{3}

The angle $\alpha$ between the space diagonal and the base lies in the same triangle, and its projection is the base diagonal:

\tan \alpha = \frac{H}{d}

A cuboid measures 6 × 6 × 6√2 cm. Find the angle between its space diagonal and the plane of the base.

Segments inside a pyramid

A square pyramid holds two different segments that are easy to confuse:

the apothem $m$ — the height of a lateral face, standing over half a base edge;
the lateral edge $b$ — running to a base vertex, so standing over half the base diagonal.

The height H, the apothem m and the base diagonal d. The angle β is the one the lateral edge makes with the base.

Both segments are hypotenuses of triangles sharing the leg $H$ and differing in the other leg:

m^2 = H^2 + \left(\frac{a}{2}\right)^2, \qquad b^2 = H^2 + \left(\frac{a\sqrt{2}}{2}\right)^2

The difference is exactly this: the apothem stands on an edge, the lateral edge on a diagonal, which is why only the second formula carries a $\sqrt{2}$ .

A square pyramid has a base edge of 12 cm and a height of 8 cm. Find its apothem and lateral edge.

Sections

A section is the figure obtained by cutting a solid with a plane. For a cylinder and a cone the important one is the axial section, through the axis of the solid: for a cylinder a $2r \times H$ rectangle, for a cone an isosceles triangle with base $2r$ and legs equal to the slant height.

A cube has two sections worth knowing:

The diagonal section of a cube runs through two opposite vertical edges. It is a rectangle with sides a and a√2.

The diagonal section is a rectangle with sides $a$ and $a\sqrt{2}$ , so its area is

P = a \cdot a\sqrt{2} = a^2\sqrt{2}

The section through the three vertices adjacent to one corner is an equilateral triangle: each of its sides is a face diagonal, that is $a\sqrt{2}$ . Its area comes from the equilateral-triangle formula:

P = \frac{(a\sqrt{2})^2\sqrt{3}}{4} = \frac{2a^2\sqrt{3}}{4} = \frac{a^2\sqrt{3}}{2}

The section through the three vertices adjacent to one corner. Every side of that triangle is a face diagonal, so the triangle is equilateral.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

diagonal of a cuboid: a = 2 cm, b = 3 cm, c = 6 cm

Common mistakes

Confusing the apothem with the lateral edge — the apothem stands over half an edge, the lateral edge over half the diagonal.
Adding only two squares for the space diagonal — all three dimensions are needed: $D^2 = a^2 + b^2 + H^2$ .
Measuring the angle to an edge instead of to the projection — the angle with a plane is taken against the orthogonal projection.
Calling skew lines parallel — having no common point is not enough; a common plane is needed too.
Assuming the diagonal section of a cube is a square — its sides are $a$ and $a\sqrt{2}$ , so it is a rectangle.
Computing the apothem from the whole base edge — the triangle contains half of it.

Formula card

Topic: Solid geometry: angles, segments and sections

Diagonal of a cuboid
$D^2 = a^2 + b^2 + H^2$
a and b are the base edges, H the height
Diagonal of a cube
$D = a\sqrt{3}$
the case a = b = c
Angle between the diagonal and the base
$\tan \alpha = \frac{H}{d}$
d is the diagonal of the base
Apothem of a pyramid
$m^2 = H^2 + \left(\frac{a}{2}\right)^2$
it stands over half a base edge
Lateral edge of a pyramid
$b^2 = H^2 + \left(\frac{a\sqrt{2}}{2}\right)^2$
it stands over half the base diagonal
Diagonal section of a cube
$P = a^2\sqrt{2}$
a rectangle with sides a and a√2

A cuboid with the base diagonal d, the space diagonal D and the angle α between them.

A square pyramid: the apothem m stands over half a base edge, while the lateral edge stands over half the base diagonal.

Frequently asked questions

A luminous right triangle with squares built on its sides drawn as a constellation, beside a circle with its radius, against a cosmic nebula in violet and magenta

Geometry

Why we measure area in squares

Perimeter is counted in metres, area in square metres — and that single exponent is behind most of the mistakes people make in plane geometry. It is why 1 m² is not 100 cm², why two plots with identical area can need wildly different amounts of fencing, and why the Pythagorean theorem is really a statement about the areas of squares rather than the lengths of sides.

11 min