Altitude of a Pyramid

Altitude of a Pyramid
Height of a Pyramid

The distance from the apex to the base of a pyramid. Formally, the shortest line segment between the apex of a pyramid and the (possibly extended) base. Altitude also refers to the length of this segment.

Key Formula

V = \frac{1}{3} B h

Where:

$V$ = Volume of the pyramid
$B$ = Area of the base
$h$ = Altitude (perpendicular height) of the pyramid

Worked Example

Problem: A right pyramid has a square base with side length 6 cm. Its volume is 84 cm³. Find the altitude of the pyramid.

Step 1: Find the area of the square base.

B = 6^2 = 36 \text{ cm}^2

Step 2: Write the volume formula and substitute the known values.

84 = \frac{1}{3}(36)(h)

Step 3: Simplify the right side.

84 = 12h

Step 4: Solve for the altitude h.

h = \frac{84}{12} = 7 \text{ cm}

Answer: The altitude of the pyramid is 7 cm.

Another Example

Problem: A right pyramid has a square base with side length 10 cm and a slant height (distance from the apex to the midpoint of a base edge) of 13 cm. Find the altitude.

Step 1: In a right pyramid with a square base, the foot of the altitude is the center of the base. The distance from the center of the square to the midpoint of an edge is half the side length.

d = \frac{10}{2} = 5 \text{ cm}

Step 2: The altitude, the distance d, and the slant height form a right triangle. Use the Pythagorean theorem, where the slant height is the hypotenuse.

13^2 = h^2 + 5^2

Step 3: Solve for h.

169 = h^2 + 25 \implies h^2 = 144 \implies h = 12 \text{ cm}

Answer: The altitude of the pyramid is 12 cm.

Frequently Asked Questions

What is the difference between the altitude and the slant height of a pyramid?

The altitude is the perpendicular distance from the apex straight down to the base plane — it goes through the interior of the pyramid. The slant height runs along a face, from the apex down to the midpoint of a base edge (or along the center of a triangular face). The slant height is always longer than or equal to the altitude.

Where does the altitude meet the base in an oblique pyramid?

In an oblique pyramid, the apex is not centered above the base, so the foot of the altitude may land outside the base region. The altitude is still measured perpendicularly to the base plane, but you may need to extend the base plane to find where it meets. The volume formula still uses this perpendicular height.

Altitude (height) vs. Slant height

The altitude is the perpendicular segment from the apex to the base plane, passing through the pyramid's interior. The slant height lies along a lateral face, running from the apex to the midpoint of a base edge. In a right pyramid, these two measurements and half a base edge form a right triangle. The altitude is always shorter than or equal to the slant height.

Why It Matters

The altitude is essential for calculating a pyramid's volume, since the formula

V = \frac{1}{3}Bh

requires the perpendicular height, not the slant height. It also appears in real-world contexts from architecture to engineering — the Great Pyramid of Giza, for instance, has an altitude of about 139 meters, which determines how much stone was needed to build it. Understanding altitude helps you connect 2D cross-sections to 3D measurements.

Common Mistakes

Mistake: Using the slant height instead of the altitude in the volume formula.

Correction: The volume formula requires the perpendicular height from the apex to the base plane. If you are given the slant height, use the Pythagorean theorem to find the altitude first.

Mistake: Assuming the altitude always passes through the center of the base.

Correction: This is only true for right pyramids. In an oblique pyramid, the apex is offset, so the foot of the altitude does not coincide with the center of the base — and it may even fall outside the base entirely.

Related Terms

Apex — Top vertex from which the altitude is drawn
Base — The face to which the altitude is perpendicular
Pyramid — The solid whose height the altitude measures
Right Pyramid — Altitude falls at the center of the base
Oblique Pyramid — Altitude foot is offset from the base center
Volume — Calculated using the altitude in V = (1/3)Bh
Altitude — General concept of perpendicular height
Surface Area — Uses slant height, often confused with altitude