What is the difference between a trapezoid and a trapezium?

In the United States, a trapezoid has exactly one pair of parallel sides; in the UK and some other countries, a trapezium refers to the same shape. The area formula A = \frac{1}{2}(b_1 + b_2)h works for both definitions, as long as you correctly identify the two parallel sides.

Can I use the trapezoid area formula if the height is a negative number?

No. Height is a distance, which is always positive by definition. If you calculate a negative value for h, you have made a sign error somewhere in your setup or calculation; recalculate carefully.

Why is there a \frac{1}{2} factor in the trapezoid area formula?

A trapezoid can be viewed as half of a parallelogram with the same two base lengths and the same perpendicular height, so the factor of \frac{1}{2} accounts for this geometric relationship.

Do the two parallel bases have to be horizontal?

No. The bases must be the two parallel sides, but they can be oriented in any direction—horizontal, vertical, or diagonal—as long as you consistently measure h perpendicular to both of them.

Area of a Trapezoid

Calculates the exact area of any trapezoid when you know the lengths of both parallel bases and the perpendicular height between them.

A = \frac{1}{2}(b_1 + b_2)h

Solve a problem with Area of a Trapezoid

Type the problem. The solver will use Area of a Trapezoid where Area of a Trapezoid is the right tool, and tell you when it is not.

What each symbol means

What Area of a Trapezoid takes

Symbol	Meaning
$A$	The area of the trapezoid, measured in square units such as $\text{cm}^2$ or $\text{m}^2$ ; do not confuse this with the perimeter.
$b_1$	The length of one of the two parallel sides (bases), measured in length units; typically called the 'bottom' base in standard orientation.
$b_2$	The length of the other parallel side (base), measured in the same length units as $b_1$ ; typically called the 'top' base in standard orientation.
$h$	The perpendicular distance (straight up at 90°) between the two parallel bases, measured in length units; this is NOT the slant height of the legs.

When to use it

Reach for this formula anytime you encounter a quadrilateral with exactly one pair of parallel sides and you know the lengths of those sides and the perpendicular distance between them.

Level

Usually taught in: Geometry · Appears on: ACT

Worked examples

1. Find the area with whole number bases and height

Problem

Find the area of a trapezoid with bases 4 cm and 6 cm and height 5 cm.

$A = \frac{1}{2}(b_1 + b_2)h$
Start with the trapezoid area formula.
$A = \frac{1}{2}(4 + 6) \cdot 5$
Substitute $b_1 = 4$ , $b_2 = 6$ , and $h = 5$ into the formula.
$A = \frac{1}{2} \cdot 10 \cdot 5$
Add the two bases: $4 + 6 = 10$ .
$A = 5 \cdot 5$
Simplify $\frac{1}{2} \cdot 10 = 5$ .
$A = 25$
Multiply $5 \cdot 5 = 25$ to find the area.

Answer: $A = 25 \text{ cm}^2$

We substituted the given values directly into the trapezoid area formula and simplified step by step, combining the bases first before multiplying by the height.

2. Find the area with decimal bases

Problem

Find the area of a trapezoid with bases 3.5 inches and 7.5 inches and height 4 inches.

$A = \frac{1}{2}(b_1 + b_2)h$
Write the trapezoid area formula.
$A = \frac{1}{2}(3.5 + 7.5) \cdot 4$
Substitute $b_1 = 3.5$ , $b_2 = 7.5$ , and $h = 4$ into the formula.
$A = \frac{1}{2} \cdot 11 \cdot 4$
Add the bases: $3.5 + 7.5 = 11$ .
$A = \frac{1}{2} \cdot 44$
Multiply the base sum by height: $11 \cdot 4 = 44$ .
$A = 22$
Divide by 2: $\frac{44}{2} = 22$ .

Answer: $A = 22 \text{ in}^2$

When the bases are decimals, add them first to get their sum before applying the rest of the formula. This prevents arithmetic errors that arise from trying to multiply mixed decimal values.

3. Find the barrier area in a tower defense game

Problem

In a tower defense game, a defensive barrier forms a trapezoid shape. The top edge (closer to enemy spawn) is 8 meters, the bottom edge (near your base) is 14 meters, and the perpendicular distance between them is 6 meters. What is the area of the barrier?

$A = \frac{1}{2}(b_1 + b_2)h$
Start with the trapezoid area formula.
$A = \frac{1}{2}(8 + 14) \cdot 6$
Substitute $b_1 = 8$ , $b_2 = 14$ , and $h = 6$ into the formula.
$A = \frac{1}{2} \cdot 22 \cdot 6$
Add the two parallel edges: $8 + 14 = 22$ .
$A = 11 \cdot 6$
Simplify $\frac{1}{2} \cdot 22 = 11$ .
$A = 66$
Multiply to find the barrier area: $11 \cdot 6 = 66$ .

Answer: $A = 66 \text{ m}^2$

Real-world scenarios like game design often present trapezoids, and this formula directly gives the coverage area of the defensive barrier, which might affect how many enemies can be blocked or the effectiveness of the defense.

Common mistakes

Where Area of a Trapezoid usually goes wrong

The mistake	Why it is wrong	The fix
Writing $A = (b_1 + b_2)h$ and forgetting the $\frac{1}{2}$ factor.	The formula requires dividing by 2 because a trapezoid with bases $b_1$ and $b_2$ has exactly half the area of a parallelogram with those same base lengths and height.	Always include the $\frac{1}{2}$ factor: a trapezoid is geometrically half of a parallelogram in terms of area.
Measuring or using the length of a slanted side leg instead of the perpendicular distance for $h$ .	The slant height is always longer than the perpendicular height (unless the trapezoid is a rectangle), so this inflates the area calculation significantly and produces an incorrect result.	The height $h$ must be perpendicular (at a 90-degree angle) to both bases. If you only know the slant height and base lengths, use the Pythagorean Theorem to find the true perpendicular height first.
Accidentally adding two heights instead of the two bases, writing something like $A = \frac{1}{2}(h_1 + h_2) \cdot b$ .	A trapezoid has exactly one height (the perpendicular distance between the parallel sides), not two; adding heights makes no geometric sense.	Remember that a trapezoid always has two bases and one height: always add the two base lengths, never add heights.

Tips and when to use something else

If the trapezoid is actually a rectangle (both bases equal), use the Area of a Rectangle formula instead; it avoids the $\frac{1}{2}$ factor and is faster.
Always label the two parallel sides as $b_1$ and $b_2$ , even if they are not the traditional 'top' and 'bottom'; this visual discipline prevents confusion about which sides to add.
If the perpendicular height is not given directly, use the Pythagorean Theorem with the slant sides and the difference between the base lengths to calculate it.
Remember: (length unit) × (length unit) = (length unit)², so your final answer must always include a squared unit label like $\text{cm}^2$ or $\text{m}^2$ .

Related formulas

Area of a TriangleCalculates the area enclosed by a triangle given its base and perpendicular height; essential for geometry, SAT, and ACT math problems involving triangles.Area of a RectangleMultiply the length and width of a rectangle to find the total area of flat space it covers, measured in square units.Area of a ParallelogramFind the area of any parallelogram by multiplying its base by the perpendicular height—a formula that works even when sides aren't perpendicular.Pythagorean TheoremFind the length of any side of a right triangle given the other two sides, or check whether a triangle is a right triangle.

Frequently asked questions

What is the difference between a trapezoid and a trapezium?: In the United States, a trapezoid has exactly one pair of parallel sides; in the UK and some other countries, a trapezium refers to the same shape. The area formula $A = \frac{1}{2}(b_1 + b_2)h$ works for both definitions, as long as you correctly identify the two parallel sides.
Can I use the trapezoid area formula if the height is a negative number?: No. Height is a distance, which is always positive by definition. If you calculate a negative value for $h$ , you have made a sign error somewhere in your setup or calculation; recalculate carefully.
Why is there a $\frac{1}{2}$ factor in the trapezoid area formula?: A trapezoid can be viewed as half of a parallelogram with the same two base lengths and the same perpendicular height, so the factor of $\frac{1}{2}$ accounts for this geometric relationship.
Do the two parallel bases have to be horizontal?: No. The bases must be the two parallel sides, but they can be oriented in any direction—horizontal, vertical, or diagonal—as long as you consistently measure $h$ perpendicular to both of them.

Need a different method?

The full solver is not scoped to one formula — type any problem and it will pick the method.

Open the math solver

Reviewed 2026-09-18