Area · Lesson 9 of 12
Area of a Trapezium
“Derive the trapezium area formula by decomposition, identify the correct height, interpret the formula using the average of the parallel sides, and solve applications.”
• Identify the two parallel sides and perpendicular height of a trapezium. • Decompose a trapezium into a rectangle and triangles. • Derive A = ½h(a + b) step by step. • Interpret the formula as average parallel-side length multiplied by height. • Apply the formula to standard and slanted trapeziums.
A trapezium has one pair of parallel sides, often of different lengths. Its area formula may look less obvious than the formulas for rectangles or parallelograms, but it can be built from shapes we already understand. The key is to drop perpendiculars and split the trapezium into a rectangle and triangles.
A quadrilateral with one pair of parallel sides. In the area formula, those parallel sides are usually represented by a and b.
Identifying the Height
The height of a trapezium is the perpendicular distance between the two parallel lines containing its parallel sides. A sloping non-parallel side is not the height unless it happens to be perpendicular to the bases.
Because the parallel sides remain a constant perpendicular distance apart, the height can be drawn at many positions. All such perpendicular segments have the same length.
Breaking the Trapezium Apart
Let the top parallel side have length a and the bottom parallel side have length b. Drop perpendiculars from both endpoints of the top side to the bottom line. This creates a central rectangle of width a and height h, plus two right triangles on the sides.
If the extra bottom pieces have lengths x and y, then b = x + a + y. The total area is the rectangle area ah plus triangle areas ½xh and ½yh.
Factor out h/2: A = ½h(2a + x + y). Since x + y = b − a, substitute to get A = ½h(2a + b − a) = ½h(a + b).
Problem
A trapezium has parallel sides 12 cm and 20 cm and height 7 cm. Find its area.
- 1.Add the parallel sides: 12 + 20 = 32 cm.
- 2.Multiply by the height: 32 × 7 = 224 cm².
- 3.Take half: A = 112 cm².
Why the Formula Works for Slanted Shapes
The derivation does not depend on the trapezium being symmetric. If one side leans outward strongly, one of the side pieces may be represented using an extension and subtraction rather than a simple interior triangle. The same algebra still reduces to ½h(a + b).
Problem
A trapezium has parallel sides 9 m and 15 m and perpendicular height 6 m. One non-parallel side extends outward beyond the other. Find its area.
- 1.The visual slant does not change which measurements enter the formula.
- 2.A = ½ × 6 × (9 + 15).
- 3.A = 3 × 24 = 72 m².
Average Parallel-Side Interpretation
The expression ½(a + b) is the average of the two parallel side lengths. So the trapezium behaves, for area purposes, like a rectangle whose width is the average base length and whose height is h.
Problem
A trapezium has parallel sides 8 cm and 18 cm with height 5 cm. Use the average-side interpretation.
- 1.Average parallel-side length = (8 + 18) ÷ 2 = 13 cm.
- 2.Area = average length × height = 13 × 5.
- 3.Area = 65 cm².
Finding an Unknown Measurement
If area and both parallel sides are known, solve for height using h = 2A/(a + b). If area, height, and one parallel side are known, solve for the other side.
Problem
A trapezium has area 150 cm² and parallel sides 10 cm and 20 cm. Find its height.
- 1.150 = ½ × h × (10 + 20).
- 2.150 = 15h.
- 3.h = 10 cm.
Connection to Other Shapes
If a = b, the trapezium formula becomes ½h(2a) = ah, exactly the area of a parallelogram. This is a useful consistency check: the general formula reduces to a familiar special case when the two parallel sides have equal length.
The formula uses a + b where a and b are the parallel sides. Adding a sloping side instead gives no valid area formula.
Quiz
Which sides are added in the trapezium area formula?
What is the height of a trapezium?
A trapezium has parallel sides 6 cm and 14 cm and height 5 cm. Its area is
The factor (a + b)/2 represents
If the two parallel sides become equal, the trapezium formula reduces to
Practice Problems
- Find the area of a trapezium with parallel sides 16 cm and 24 cm and height 9 cm.
- A trapezium has area 126 cm², parallel sides 10 cm and 18 cm. Find its height.
- A trapezium has area 96 m², height 8 m, and one parallel side 10 m. Find the other parallel side.
- Explain why a sloping side should not be substituted for h.
- Decompose a trapezium into a rectangle and two triangles and write the area sum before simplifying.
- Show algebraically that when a = b, ½h(a + b) becomes bh.
Key Takeaways
• The trapezium height is the perpendicular distance between the parallel sides. • Decomposition into a rectangle and triangles leads naturally to the formula. • Area = ½h(a + b), where a and b are the parallel sides. • The same formula can be read as average parallel-side length × height. • Strong slant or asymmetry does not change the formula. • The formula reduces to parallelogram area when the two parallel sides are equal.