The trapezoid area formula and how to use it
Why the trapezoid area formula averages the parallel sides, how to find the perpendicular height, and how area differs from perimeter.
Why you average the two bases
A trapezoid has two parallel sides of different lengths, so it has no single base like a rectangle. The formula handles this by averaging them: add side a and side b, then divide by two. That average is the width of an equivalent rectangle with the same height and the same area. Multiplying the average width by the height gives the area, which is why (a + b) / 2 times height works for every trapezoid regardless of how lopsided it looks.
Using the perpendicular height, not the slant
The height in the formula is the straight up and down distance between the two parallel sides, measured at a right angle to them. It is not the length of a slanted side. This trips people up when a diagram only labels the sloping edge. If you know a slant length and its angle, take the vertical component before entering it here. Using a slant length by mistake overstates the height and inflates the area.
Area versus perimeter
Area measures the surface the trapezoid covers and needs only the two parallel sides and the height. Perimeter measures the distance around the outside and needs all four side lengths. That is why the perimeter option asks for the two slanted legs, c and d, on top of the parallel sides. Adding a, b, c and d gives the perimeter, so a shape with sides 6, 10, 5 and 7 has a perimeter of 28 even though its area depends on the height instead of those legs.
A quick worked example
Suppose the parallel sides are 6 and 10 units and the perpendicular height is 4 units. Average the bases to get 8, then multiply by the height of 4 to reach an area of 32 square units. If the two slanted sides measure 5 and 7, the perimeter is 6 plus 10 plus 5 plus 7, which is 28 units. The calculator performs both steps at once when you enable the perimeter option, so you can check work like this in a single glance.