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Altitude

Altitude
Height

The shortest distance between the base of a geometric figure and its top, whether that top is an opposite vertex, an apex, or another base.

 

 

See also

Altitude of a cone, altitude of a cylinder, altitude of a parallelogram, altitude of a prism, altitude of a pyramid, altitude of a trapezoid, altitude of a triangle

Key Formula

A=12 b hA = \tfrac{1}{2}\,b\,h
Where:
  • AA = Area of a triangle
  • bb = Length of the base
  • hh = Altitude (perpendicular height) drawn to that base

Worked Example

Problem: A triangle has a base of 10 cm and an area of 30 cm². Find its altitude to that base.
Step 1: Write the area formula for a triangle.
A=12 b hA = \tfrac{1}{2}\,b\,h
Step 2: Substitute the known values.
30=12(10)(h)30 = \tfrac{1}{2}(10)(h)
Step 3: Simplify the right side.
30=5h30 = 5h
Step 4: Solve for the altitude by dividing both sides by 5.
h=305=6h = \frac{30}{5} = 6
Answer: The altitude of the triangle is 6 cm.

Another Example

Problem: A trapezoid has parallel bases of length 8 cm and 12 cm, and its area is 60 cm². Find the altitude (the perpendicular distance between the two bases).
Step 1: Write the area formula for a trapezoid.
A=12(b1+b2) hA = \tfrac{1}{2}(b_1 + b_2)\,h
Step 2: Substitute the known values.
60=12(8+12) h60 = \tfrac{1}{2}(8 + 12)\,h
Step 3: Simplify inside the parentheses and multiply.
60=12(20) h=10h60 = \tfrac{1}{2}(20)\,h = 10h
Step 4: Solve for h.
h=6010=6h = \frac{60}{10} = 6
Answer: The altitude of the trapezoid is 6 cm.

Frequently Asked Questions

Is the altitude of a triangle always inside the triangle?
No. In an obtuse triangle, the altitude drawn from an acute-angle vertex lands outside the triangle because the perpendicular from that vertex to the opposite side (extended) falls beyond the base. The altitude from the vertex of the obtuse angle, however, does fall inside. So the location depends on which vertex you draw from and the triangle's shape.
What is the difference between altitude and height?
In geometry, altitude and height mean the same thing: the perpendicular distance from the base to the top of a figure. The word "altitude" is more formal and is also used to refer to the line segment itself, while "height" is the everyday term for its length.

Altitude vs. Median (of a triangle)

An altitude drops perpendicularly from a vertex to the opposite side (or its extension). A median connects a vertex to the midpoint of the opposite side. A median bisects the opposite side; an altitude does not, unless the triangle is isosceles with respect to that vertex. They coincide only in an equilateral triangle or along the axis of symmetry of an isosceles triangle.

Why It Matters

Nearly every area and volume formula in geometry depends on knowing the altitude. For triangles, parallelograms, trapezoids, prisms, pyramids, cylinders, and cones, the altitude is the perpendicular height that plugs directly into the formula. Understanding altitude also connects to coordinate geometry and trigonometry, where you compute perpendicular distances to solve real-world problems in engineering, architecture, and physics.

Common Mistakes

Mistake: Using a slant side of the figure instead of the perpendicular distance as the altitude.
Correction: The altitude must form a 90° angle with the base. A slant height or a non-perpendicular side is longer than the true altitude. Always look for (or construct) the right-angle measurement.
Mistake: Assuming the altitude of a triangle always lies inside the triangle.
Correction: For obtuse triangles, one or two altitudes fall outside the triangle. You may need to extend the base line to find where the perpendicular from the opposite vertex meets it.

Related Terms