Geometry Reference

Trapezoids

A trapezoid is a quadrilateral with one pair of parallel sides. Most trapezoid problems use the bases, height, median, or the special angle and diagonal facts of isosceles trapezoids.

Solved Examples

Jump directly to a worked example by trapezoid task.

Summary Table

Feature Meaning Rule
Bases The parallel sides. Usually called $b_1$ and $b_2$.
Legs The non-parallel sides. Congruent legs make an isosceles trapezoid.
Height Perpendicular distance between the bases. Use perpendicular height, not a slanted leg.
Median or midsegment Segment joining the midpoints of the legs. Parallel to the bases and equal to their average.
Same-leg interior angles Angles along one leg between parallel bases. They are supplementary.
Isosceles trapezoid Legs are congruent. Base angles are congruent and diagonals are congruent.

Content Formulas

Area
$$A=\frac12(b_1+b_2)h$$
Median / Midsegment
$$m=\frac{b_1+b_2}{2}$$
Area with Median
$$A=mh$$
Supplementary Same-Leg Angles
$$\angle A+\angle D=180^\circ$$ $$\angle B+\angle C=180^\circ$$
The area formula can be read as average of the bases times height. That average is exactly the median length.

Classic Mastery Problems

Area from Bases and Height

A trapezoid has bases $8$ and $14$ and height $5$. Find its area.

Solution Steps
  1. Average the bases and multiply by the perpendicular height.
  2. Substitute the given values.
  3. State the area: $55$ square units.
$$A=\frac12(b_1+b_2)h$$ $$=\frac12(8+14)(5)$$ $$=55$$

Find a Missing Base

A trapezoid has area $96$, height $8$, and one base $10$. Find the other base.

Solution Steps
  1. Substitute the known area, height, and base.
  2. Simplify the product.
  3. Isolate the missing base.
  4. State the missing base: $14$.
$$96=\frac12(10+b_2)(8)$$ $$96=4(10+b_2)$$ $$24=10+b_2$$ $$b_2=14$$

Median Length

The bases of a trapezoid are $11$ and $25$. Find the median length.

Solution Steps
  1. Average the two bases.
  2. State the median length: $18$.
$$m=\frac{11+25}{2}$$ $$=18$$

Use Median to Find Area

A trapezoid has median $17$ and height $9$. Find its area.

Solution Steps
  1. Use area equals median times height.
  2. Substitute the median and height.
  3. State the area: $153$ square units.
$$A=mh$$ $$=17(9)$$ $$=153$$

Same-Leg Angles

In a trapezoid, one angle along a leg is $68^\circ$. Find the other angle along the same leg.

Solution Steps
  1. Use supplementary same-leg interior angles.
  2. State the other angle: $112^\circ$.
$$68^\circ+x=180^\circ$$ $$x=112^\circ$$

Isosceles Trapezoid Angles

An isosceles trapezoid has one base angle of $74^\circ$. Find the other three angles.

Solution Steps
  1. Use congruent base angles.
  2. Subtract the given angle from $180^\circ$.
  3. State the other three angles.
$$\text{same base angle: }74^\circ$$ $$180^\circ-74^\circ=106^\circ$$ $$\text{other angles: }74^\circ,\ 106^\circ,\ 106^\circ$$

Isosceles Trapezoid Diagonals

In an isosceles trapezoid, the diagonals are labeled $3x+2$ and $5x-10$. Find $x$ and the diagonal length.

Solution Steps
  1. Set the congruent diagonal expressions equal.
  2. Isolate $x$.
  3. State $x=6$.
  4. Substitute $x$ to find the diagonal length.
$$3x+2=5x-10$$ $$12=2x$$ $$x=6$$ $$3(6)+2=20$$

Trapezoid Checklist

Question Check
Which sides are the bases? Find the parallel sides.
Is the height given? Use perpendicular height, not a leg.
Is a median or midsegment given? Treat it as the average of the bases.
Is the trapezoid isosceles? Use congruent legs, congruent base angles, and congruent diagonals.
Are angles on the same leg? They add to $180^\circ$.