How is an obelisk different from a frustum of a pyramid?

A frustum is created by slicing a pyramid with a plane parallel to its base, so the top and base must be similar rectangles (same length-to-width ratio). An obelisk allows the top rectangle to have a completely different ratio from the base. For example, the base could be 6 × 4 while the top is 3 × 1 — something impossible for a true frustum. Every rectangular frustum is an obelisk, but not every obelisk is a frustum.

Obelisk — Definition, Formula & Examples

An obelisk is a three-dimensional solid that has a rectangular base, a smaller rectangular top parallel to the base, and four trapezoidal faces connecting them. Unlike a frustum of a pyramid, the base and top rectangles do not need to be similar — they can have different length-to-width ratios.

An obelisk is a prismatoid whose two parallel faces (the base and top) are rectangles with dimensions $a \times b$ and $c \times d$ respectively, where the centers of the two rectangles are aligned along a perpendicular axis of height $h$ . Each lateral face is a trapezoid. When $c = d = 0$ , the obelisk reduces to a rectangular pyramid; when $a = c$ and $b = d$ , it becomes a rectangular prism.

Key Formula

V = \frac{h}{6}\bigl(ab + cd + (a+c)(b+d)\bigr)

Where:

$V$ = Volume of the obelisk
$a$ = Length of the base rectangle
$b$ = Width of the base rectangle
$c$ = Length of the top rectangle
$d$ = Width of the top rectangle
$h$ = Perpendicular height between the base and top

How It Works

To find the volume of an obelisk, you use a formula derived from the prismoidal formula, which generalizes volume calculations for solids bounded by two parallel planes. You need five measurements: the length

a

and width

b

of the base rectangle, the length

c

and width

d

of the top rectangle, and the perpendicular height

h

between them. The formula accounts for the way the cross-section changes from one rectangle to the other. This shape appears in architecture, monument design, and excavation calculations where a trench tapers from a wider bottom to a narrower top.

Worked Example

Problem: Find the volume of an obelisk with a base of 6 m × 4 m, a top of 3 m × 2 m, and a height of 9 m.

Identify measurements: Base: a = 6, b = 4. Top: c = 3, d = 2. Height: h = 9.

Compute each product in the formula: Find ab, cd, and (a+c)(b+d).

ab = 6 \times 4 = 24, \quad cd = 3 \times 2 = 6, \quad (a+c)(b+d) = 9 \times 6 = 54

Sum the three terms: Add the three results together.

24 + 6 + 54 = 84

Multiply by h/6: Apply the leading factor.

V = \frac{9}{6} \times 84 = 1.5 \times 84 = 126 \text{ m}^3

Answer: The volume of the obelisk is 126 m³.

Another Example

Problem: Verify that the obelisk formula gives the correct volume for a rectangular pyramid with base 10 m × 8 m and height 12 m (top dimensions c = 0 and d = 0).

Set up the formula: With c = 0 and d = 0, the formula simplifies.

V = \frac{h}{6}\bigl(ab + 0 + (a+0)(b+0)\bigr) = \frac{h}{6}(ab + ab) = \frac{h}{6}(2ab) = \frac{hab}{3}

Substitute values: Plug in a = 10, b = 8, h = 12.

V = \frac{12 \times 10 \times 8}{3} = \frac{960}{3} = 320 \text{ m}^3

Check against pyramid formula: The standard pyramid volume is (1/3) × base area × height.

V = \frac{1}{3}(80)(12) = 320 \text{ m}^3 \checkmark

Answer: Both formulas give 320 m³, confirming the obelisk formula reduces correctly to the pyramid formula.

Why It Matters

The obelisk formula is used in civil engineering and construction to calculate earthwork volumes when trenches, embankments, or foundations taper from one rectangular cross-section to another. In high-school and college solid geometry courses, it demonstrates how the prismoidal formula unifies volume calculations for prisms, pyramids, and frustums under a single framework.

Common Mistakes

Mistake: Using the frustum formula when the base and top rectangles are not similar.

Correction: The frustum formula assumes similar cross-sections. When the top and base rectangles have different length-to-width ratios, you must use the obelisk (prismoidal) formula instead.

Mistake: Forgetting the mixed term (a+c)(b+d) and averaging only the two rectangular areas.

Correction: A simple average of the top and base areas ignores how the shape transitions between them. The middle term (a+c)(b+d) accounts for the cross-sectional area at the midpoint and is essential for accuracy.

Related Terms

Altitude of a Pyramid — Height measured the same way as in an obelisk
Altitude of a Prism — An obelisk with equal top and base is a prism
Base — The bottom rectangular face of an obelisk
Apex — An obelisk with zero top dimensions has an apex
Cavalieri's Principle — Foundational principle for deriving volume formulas
Altitude of a Cone — Analogous height concept for circular solids