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Altitude of a Cone

Altitude of a Cone
Height of a Cone

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

 

Cone with a dashed vertical line labeled "altitude" from apex to center of circular base.

 

 

See also

Oblique cone, right cone, volume, surface area

Key Formula

V=13πr2hV = \frac{1}{3}\pi r^2 h
Where:
  • VV = Volume of the cone
  • rr = Radius of the circular base
  • hh = Altitude (height) of the cone — the perpendicular distance from apex to base
  • π\pi = Pi, approximately 3.14159

Worked Example

Problem: A right cone has a slant height of 13 cm and a base radius of 5 cm. Find the altitude of the cone.
Step 1: In a right cone, the altitude, the radius, and the slant height form a right triangle. The slant height is the hypotenuse.
l2=r2+h2l^2 = r^2 + h^2
Step 2: Substitute the known values: slant height l = 13 cm and radius r = 5 cm.
132=52+h213^2 = 5^2 + h^2
Step 3: Simplify and solve for h².
169=25+h2  ⟹  h2=144169 = 25 + h^2 \implies h^2 = 144
Step 4: Take the positive square root to find the altitude.
h=144=12 cmh = \sqrt{144} = 12 \text{ cm}
Answer: The altitude of the cone is 12 cm.

Another Example

Problem: A cone has an altitude of 9 cm and a base radius of 4 cm. Find its volume.
Step 1: Write the volume formula for a cone.
V=13πr2hV = \frac{1}{3}\pi r^2 h
Step 2: Substitute r = 4 cm and h = 9 cm.
V=13π(4)2(9)=13π(16)(9)V = \frac{1}{3}\pi (4)^2 (9) = \frac{1}{3}\pi (16)(9)
Step 3: Compute the result.
V=144π3=48π≈150.8 cm3V = \frac{144\pi}{3} = 48\pi \approx 150.8 \text{ cm}^3
Answer: The volume of the cone is 48π ≈ 150.8 cm³.

Frequently Asked Questions

What is the difference between the altitude and the slant height of a cone?
The altitude is the perpendicular distance from the apex straight down to the base plane. The slant height is the distance measured along the surface of the cone from the apex to the edge of the base. In a right cone, the altitude is shorter than the slant height because the slant height is the hypotenuse of the right triangle they form with the radius.
How do you find the altitude of a cone if you know the volume and radius?
Rearrange the volume formula: h = 3V / (πr²). Multiply the volume by 3, then divide by π times the radius squared. For example, if V = 48π cm³ and r = 4 cm, then h = 3(48π) / (π · 16) = 144π / 16π = 9 cm.

Altitude (height) vs. Slant height

The altitude of a cone is an internal measurement — the perpendicular distance from the apex to the base plane. The slant height runs along the lateral surface from the apex to the base circle. For a right cone with altitude h, radius r, and slant height l, these are related by the Pythagorean theorem: l² = r² + h². The altitude is always less than or equal to the slant height.

Why It Matters

The altitude appears in every major cone formula. You need it to calculate volume (V=13πr2hV = \frac{1}{3}\pi r^2 h) and to find the slant height, which is required for lateral surface area. Understanding that altitude is always perpendicular to the base — even in oblique cones where the apex is not directly above the center — prevents errors in geometry and real-world applications like engineering and architecture.

Common Mistakes

Mistake: Confusing the altitude with the slant height and using the slant height in the volume formula.
Correction: The volume formula requires the altitude (the perpendicular height), not the slant height. The slant height is always longer than the altitude in a non-degenerate cone, so using it would give an incorrect, larger volume.
Mistake: Assuming the altitude always passes through the center of the base.
Correction: This is true only for a right cone. In an oblique cone, the apex is off-center, so the altitude drops perpendicularly to the base plane but does not hit the center of the circular base.

Related Terms

  • Cone — The solid whose altitude is being measured
  • Apex — Top point where the altitude starts
  • Base — The flat face the altitude is perpendicular to
  • Right Cone — Cone whose altitude passes through base center
  • Oblique Cone — Cone whose altitude does not hit base center
  • Volume — Calculated using the altitude in the formula
  • Surface Area — Uses slant height derived from the altitude
  • Altitude — General concept of perpendicular height