Median of a Trapezoid — Definition, Formula & Examples

Median of a Trapezoid

A line segment connecting the midpoints of the legs of a trapezoid. The median is parallel to the bases. Note: Its length equals the arithmetic mean of the lengths of the bases.

Trapezoid with a horizontal dashed line labeled "median" connecting the midpoints of the two non-parallel sides (legs).

Key Formula

m = \frac{b_1 + b_2}{2}

Where:

$m$ = Length of the median (midsegment) of the trapezoid
$b_1$ = Length of one base (one of the two parallel sides)
$b_2$ = Length of the other base

Worked Example

Problem: A trapezoid has bases of length 10 cm and 16 cm. Find the length of the median.

Step 1: Identify the two base lengths.

b_1 = 10 \text{ cm}, \quad b_2 = 16 \text{ cm}

Step 2: Write the median formula.

m = \frac{b_1 + b_2}{2}

Step 3: Substitute the values and compute.

m = \frac{10 + 16}{2} = \frac{26}{2} = 13 \text{ cm}

Answer: The median of the trapezoid is 13 cm.

Another Example

This example works the formula in reverse — solving for an unknown base when the median and one base are given, which is a common exam variation.

Problem: The median of a trapezoid is 20 cm long, and one base measures 14 cm. Find the length of the other base.

Step 1: Write down what you know: the median and one base.

m = 20 \text{ cm}, \quad b_1 = 14 \text{ cm}

Step 2: Set up the median formula and solve for the unknown base.

20 = \frac{14 + b_2}{2}

Step 3: Multiply both sides by 2.

40 = 14 + b_2

Step 4: Subtract 14 from both sides.

b_2 = 40 - 14 = 26 \text{ cm}

Answer: The other base is 26 cm.

Frequently Asked Questions

What is the difference between the median of a trapezoid and the median of a triangle?

The median of a trapezoid connects the midpoints of the two legs (non-parallel sides) and is parallel to the bases. The median of a triangle connects a vertex to the midpoint of the opposite side. Despite sharing the name 'median,' they connect different types of points and satisfy different length formulas.

Is the median of a trapezoid the same as the midsegment?

Yes. Many textbooks call this segment the midsegment of a trapezoid instead of the median. Both terms refer to the same segment — the one joining the midpoints of the two legs. The formula is the same regardless of which name is used.

Why is the median of a trapezoid parallel to the bases?

By the triangle midsegment theorem applied to the diagonals of the trapezoid, the segment joining the midpoints of the legs must be parallel to both bases. You can also verify this using coordinate geometry: placing the trapezoid on a coordinate plane shows the median has the same slope as both bases.

Median of a Trapezoid vs. Median of a Triangle

	Median of a Trapezoid	Median of a Triangle
Definition	Segment connecting the midpoints of the two legs of a trapezoid	Segment connecting a vertex to the midpoint of the opposite side of a triangle
Formula	m = (b₁ + b₂) / 2	No simple average formula; length depends on all three sides
Number per shape	Exactly one median per trapezoid	Exactly three medians per triangle (one from each vertex)
Parallelism	Always parallel to both bases	Not necessarily parallel to any side
Key property	Its length is the arithmetic mean of the two bases	All three medians meet at the centroid, which divides each median in a 2:1 ratio

Why It Matters

The median of a trapezoid appears frequently in geometry courses when you study quadrilateral properties and area calculations. Knowing the median lets you quickly find the area of a trapezoid, since the area formula

A = m \cdot h

(where

h

is the height) is equivalent to

A = \frac{(b_1 + b_2)}{2} \cdot h

. You will also encounter this concept in coordinate geometry proofs and standardized tests such as the SAT and ACT.

Common Mistakes

Mistake: Subtracting the bases instead of adding them.

Correction: The median is the average (arithmetic mean) of the two bases, so you add them and divide by 2: m = (b₁ + b₂) / 2. Subtracting would give a much smaller (or even negative) result, which doesn't make sense for a length.

Mistake: Confusing the median with the height of the trapezoid.

Correction: The median is a horizontal segment parallel to the bases, while the height is the perpendicular distance between the bases. They are different measurements and serve different roles in calculations.

Related Terms

Line Segment — The median is a specific line segment
Midpoint — Endpoints of the median are midpoints of the legs
Leg of a Trapezoid — The two non-parallel sides the median connects
Parallel Lines — The median is parallel to both bases
Base of a Trapezoid — The two parallel sides used in the median formula
Arithmetic Mean — The median length equals the mean of the bases
Median of a Triangle — A different type of median in a different shape
Median of a Set of Numbers — Same word but a statistical concept, not geometric