A trapezium has one pair of parallel sides. Enter both parallel sides and the perpendicular height to find its area.
Enter both parallel sides.
Enter the perpendicular height.
Read the area.
Area: A = ½ × (a + b) × h
A trapezium is a four-sided shape with exactly one pair of parallel sides. In American books it is called a trapezoid. This calculator takes the two parallel sides a and b and the perpendicular height h between them, and returns the area ½ × (a + b) × h.
Trapeziums are common in land records and construction in India. Many plots are wider at the road than at the back, and their area is worked out this way. The cross-section of a canal, drain, embankment or road cutting is usually a trapezium, so its area gives the volume of earth to dig per metre of length. Roof slopes, table tops, handbag panels and the side view of a bucket are other examples. The formula is taught in Class 8 mensuration.
1. Identify the two parallel sides, a and b. They are the top and bottom of the shape when it is drawn with them horizontal.
2. Measure the height h: the perpendicular distance between the parallel sides, not the length of a slanting side.
3. Add the parallel sides: a + b.
4. Take half the sum, which is their average: (a + b) ÷ 2.
5. Multiply by the height: Area = ½ × (a + b) × h.
6. State the answer in square units; for a canal or trench, multiply by its length to get the volume.
Take a copy of the trapezium, turn it upside down and place it beside the original so the slanting sides meet. The two pieces form a parallelogram whose base is a + b and whose height is h. Its area is (a + b) × h, and the trapezium is exactly half of it. Another way to see it is to split the trapezium along a diagonal into two triangles with bases a and b and the same height, giving ½ah + ½bh.
The formula can be read as average width × height. The width of a trapezium changes steadily from a to b as you go from one parallel side to the other, so its average is (a + b) ÷ 2, which is the length of the midline joining the midpoints of the slanting sides. This view links the trapezium to the trapezoidal rule in calculus and surveying, which estimates an irregular area by slicing it into strips and treating each as a trapezium.
If a = b, both pairs of sides are parallel and the shape is a parallelogram, and the formula gives a × h. If one parallel side shrinks to zero, it becomes a triangle with area ½ × b × h. The slanting sides do not appear in the formula at all: trapeziums with very different slants but the same parallel sides and height have the same area. You need the slant lengths only for the perimeter, or to find h by Pythagoras.
Gopal's plot is a trapezium with a 24 m frontage on the road and a 16 m back boundary parallel to it, and the two boundaries are 9 m apart.
Area = ½ × (a + b) × h: = 0.5 × (24 + 16) × 9 = 180
Answer: Area 180
Using the length of a slanting side as the height.
Adding the two non-parallel sides instead of the parallel ones.
Forgetting to halve the sum, which doubles the area.
Mixing units, such as parallel sides in feet and height in metres.
Treating an irregular quadrilateral with no parallel sides as a trapezium; split it into triangles instead.
Finding the area of plots that taper from the road frontage to the back.
Calculating the cross-section of canals, drains and embankments for earthwork estimates.
Estimating areas of trapezium-shaped roof panels, walls and glass sheets.
Applying the trapezoidal rule to estimate irregular areas from strip measurements.
Solving Class 8 to 10 mensuration problems.
Is the height the slanted side?
No, it is the perpendicular distance between the parallel sides.
Are trapezium and trapezoid the same?
In Indian and British usage, yes: a quadrilateral with one pair of parallel sides.