Altitude of a Prism

Q: How do you find the altitude of a prism if you know the volume and base area?

Rearrange the volume formula: $h = \frac{V}{B}$. Divide the volume by the area of one base. For example, if a prism has volume 300 cm³ and base area 50 cm², then $h = 300 \div 50 = 6$ cm.

Altitude of a Prism
Height of a Prism

The distance between the two bases of a prism. Formally, the shortest line segment between the (possibly extended) bases. Altitude also refers to the length of this segment.

Key Formula

V = B \times h

Where:

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

Worked Example

Problem: A right triangular prism has bases that are equilateral triangles with side length 6 cm. The altitude of the prism is 10 cm. Find the volume of the prism.

Step 1: Find the area of the triangular base. An equilateral triangle with side length

s

has area

\frac{s^2\sqrt{3}}{4}

B = \frac{6^2 \sqrt{3}}{4} = \frac{36\sqrt{3}}{4} = 9\sqrt{3} \approx 15.59 \text{ cm}^2

Step 2: Identify the altitude. Since this is a right prism, the altitude equals the length of the lateral edge. Here

h = 10

cm.

h = 10 \text{ cm}

Step 3: Apply the volume formula: multiply the base area by the altitude.

V = B \times h = 9\sqrt{3} \times 10 = 90\sqrt{3} \approx 155.9 \text{ cm}^3

Answer: The volume of the prism is

90\sqrt{3} \approx 155.9

cm³.

Another Example

Problem: An oblique rectangular prism has a base that is 8 cm by 5 cm. The lateral edges are each 13 cm long but are tilted so that the perpendicular distance between the two bases is only 12 cm. Find the volume.

Step 1: Find the base area.

B = 8 \times 5 = 40 \text{ cm}^2

Step 2: Identify the altitude. The lateral edge is 13 cm, but the prism is oblique, so the altitude is the perpendicular distance between the bases: 12 cm. Do not use 13 cm.

h = 12 \text{ cm}

Step 3: Compute the volume using the altitude, not the lateral edge length.

V = B \times h = 40 \times 12 = 480 \text{ cm}^3

Answer: The volume is 480 cm³. Notice that using the lateral edge (13 cm) instead of the altitude (12 cm) would give an incorrect answer of 520 cm³.

Frequently Asked Questions

Is the altitude of a prism the same as the lateral edge?

Only for a right prism. In a right prism, the lateral edges are perpendicular to the bases, so the lateral edge length equals the altitude. In an oblique prism, the lateral edges are tilted, making them longer than the altitude. The altitude is always measured perpendicular to the base planes.

How do you find the altitude of a prism if you know the volume and base area?

Rearrange the volume formula:

h = \frac{V}{B}

. Divide the volume by the area of one base. For example, if a prism has volume 300 cm³ and base area 50 cm², then

h = 300 \div 50 = 6

cm.

Right Prism vs. Oblique Prism

In a right prism, the lateral faces are perpendicular to the bases, so the altitude equals the lateral edge length. In an oblique prism, the lateral faces are tilted, so the lateral edges are longer than the altitude. The altitude is always the perpendicular distance between the two parallel base planes, regardless of the prism type. This distinction matters most when computing volume.

Why It Matters

The altitude is essential for calculating the volume of any prism. Without the correct perpendicular height, volume calculations will be wrong—especially for oblique prisms where the slant length differs from the true height. The concept also extends to cylinders and other solids with parallel bases, making it a foundational idea in solid geometry.

Common Mistakes

Mistake: Using the lateral edge length as the altitude for an oblique prism.

Correction: The altitude is the perpendicular distance between the two base planes, which is shorter than the lateral edge in an oblique prism. Only in a right prism are they the same.

Mistake: Confusing the altitude of the prism with the altitude of the base.

Correction: A triangular prism, for instance, has two different 'altitudes': the height of the triangular base (used to find base area) and the altitude of the prism itself (the distance between the two triangular faces). Keep these separate and use each in the correct part of the calculation.

Related Terms

Prism — The solid whose height the altitude measures
Right Prism — Prism where altitude equals lateral edge length
Oblique Prism — Prism where altitude is shorter than lateral edge
Base — The two parallel faces between which altitude is measured
Volume — Computed as base area times altitude
Altitude — General concept of perpendicular height
Line Segment — The altitude is a specific perpendicular segment
Surface Area — Lateral surface area also depends on prism height