Area · Lesson 10 of 12
More Ways to Understand Trapezium Area
“Prove the trapezium formula using two copies and alternative dissections, transform trapeziums to equal-area rectangles, and solve advanced area relationships.”
• Derive the trapezium formula by joining two congruent copies into a parallelogram. • Compare multiple valid derivations of the same area formula. • Transform a trapezium into an equal-area rectangle by dissection. • Construct trapeziums with a prescribed area. • Use trapezium area reasoning inside larger composite and proof-style problems.
A strong formula should survive more than one derivation. The trapezium formula can be obtained by splitting one trapezium into familiar pieces, but there is another particularly visual argument: join two identical copies to make a parallelogram. This method explains the factor one half almost immediately.
The Two-Copy Derivation
Take two congruent copies of a trapezium with parallel sides a and b and height h. Rotate the second copy and join it to the first along a non-parallel side. The supplementary interior angles along the parallel sides line up to create straight edges, and the outside boundary becomes a parallelogram.
The new parallelogram has base a + b and height h. Its area is h(a + b). Because it is made from two identical trapeziums, one trapezium has half that area.
Problem
A trapezium has parallel sides 7 cm and 13 cm and height 6 cm. Use the parallelogram formed by two copies.
- 1.Two copies form a parallelogram with base 7 + 13 = 20 cm and height 6 cm.
- 2.Parallelogram area = 20 × 6 = 120 cm².
- 3.One trapezium has half that area: 60 cm².
Alternative Decompositions
A trapezium can also be handled as a larger rectangle minus a triangle, or as a parallelogram plus or minus a triangle, depending on its orientation. These methods are not separate formulas; they are different routes to the same invariant area.
Being able to switch methods is useful when a diagram gives awkward measurements for one decomposition but convenient measurements for another.
Problem
A trapezium fits inside a 16 cm by 8 cm rectangle. The cut-off right triangle has base 6 cm and height 8 cm. Find the trapezium area.
- 1.Rectangle area = 16 × 8 = 128 cm².
- 2.Removed triangle area = ½ × 6 × 8 = 24 cm².
- 3.Trapezium area = 128 − 24 = 104 cm².
Trapezium to Equal-Area Rectangle
Since A = ½h(a + b), an equal-area rectangle can have height h and width (a + b)/2. This is the average of the parallel sides. A dissection can physically rearrange the trapezium into a rectangle with exactly those dimensions.
Problem
A trapezium has parallel sides 10 cm and 18 cm with height 7 cm. Give dimensions of an equal-area rectangle.
- 1.Average parallel-side length = (10 + 18)/2 = 14 cm.
- 2.Keep the same height 7 cm.
- 3.The 14 cm by 7 cm rectangle has area 98 cm², equal to the trapezium.
Designing a Trapezium with a Given Area
To construct a trapezium with a prescribed area, choose any positive height h and any positive parallel sides a and b satisfying ½h(a + b) = desired area. There are many possible answers, which makes this a design problem rather than a single-answer calculation.
Problem
Construct one trapezium of area 144 cm².
- 1.Choose height h = 8 cm.
- 2.Then 144 = ½ × 8 × (a + b), so a + b = 36 cm.
- 3.Choose any positive pair adding to 36, such as a = 14 cm and b = 22 cm.
- 4.A trapezium with parallel sides 14 cm and 22 cm and height 8 cm has area 144 cm².
Composite Shapes and Area Ratios
A regular hexagon can be partitioned into familiar pieces such as a trapezium, rhombus, and equilateral triangle. Rather than memorising a ratio, break each part into the same small equilateral-triangle units. The ratio of areas then becomes a ratio of equal unit-triangle counts.
This unit-piece strategy is more robust than relying on visual size. Two regions that look different may contain the same number of congruent small triangles and therefore have equal area.
A Trapezium and an Equal-Area Triangle
In some configurations, moving or extending a vertex can turn a trapezium into a triangle with the same area. A midpoint condition is often the clue: it creates equal-base or equal-area triangles that can be added to one shape and removed from another.
Problem
Suppose a trapezium and a triangle share a base, and a midpoint construction creates two small triangles with equal bases and the same height. Explain how this can prove the trapezium and larger triangle have equal area.
- 1.Identify the two small triangles created by the midpoint.
- 2.Use equal bases and common height to show those small triangles have equal areas.
- 3.Write each large region as a common central area plus one of the equal small triangles.
- 4.The totals are equal because equal areas are added to the same common area.
A pair of pieces may look as if they fit, but a dissection proof should justify matching lengths or angles. When two copies form a parallelogram, parallel-line angle relationships explain why the outer edges become straight and parallel.
Quiz
What figure do two suitably arranged congruent trapeziums form in the two-copy derivation?
What is the base of that parallelogram in terms of trapezium parallel sides?
Why is one trapezium half the parallelogram area?
An equal-area rectangle with the same height h can have width
To design a trapezium of fixed area, which statement is true?
Practice Problems
- Use the two-copy method to find the area of a trapezium with parallel sides 11 cm and 17 cm and height 9 cm.
- A trapezium has parallel sides 9 cm and 21 cm and height 6 cm. Give dimensions of an equal-area rectangle with the same height.
- Design two different trapeziums of area 120 cm².
- A trapezium sits inside a 20 cm by 10 cm rectangle with a cut-off triangle of base 8 cm and height 10 cm. Find the trapezium area.
- Explain why the two-copy construction produces a parallelogram rather than a six-sided outer boundary.
- Describe how equal small-triangle counts can be used to compare areas in a partitioned regular hexagon.
- Write a proof outline for an equal-area claim that uses a midpoint to create two equal small triangles.
Key Takeaways
• Two congruent trapeziums can be arranged into a parallelogram of area h(a + b). • One trapezium is therefore half that area. • Rectangle, triangle, parallelogram, and two-copy methods are different derivations of the same formula. • An equal-area rectangle can use width (a + b)/2 and height h. • A fixed target area can be achieved by many different trapezium dimensions. • Midpoints and equal small triangles are powerful tools in area-equality proofs.