Area · Lesson 5 of 12
Equal-Area Transformations and Geometric Dissection
“Use cutting and rearrangement to transform rectangles and triangles while preserving area, with connections to classical Indian geometric constructions.”
• Explain why cutting and rearranging pieces without gaps or overlaps preserves total area. • Define geometric dissection and use it to transform one shape into another. • Construct equal-area rectangle-to-triangle and triangle-to-rectangle transformations. • Analyse simple isosceles-triangle dissections. • Connect area-preserving constructions with geometric ideas found in the Śulba-Sūtras.
Area is attached to the pieces of a figure, not to the outline that happens to surround them. If we cut a shape into pieces and rearrange every piece without stretching, overlapping, or leaving one out, the total area stays exactly the same. The boundary may change completely.
This principle lets us prove formulas and design shapes with prescribed area. It also links modern classroom geometry with very old construction traditions in which one altar shape had to be transformed into another while preserving area.
What Is Dissection?
A method of cutting a figure into a finite number of pieces and rearranging those same pieces to form another figure. If no piece is stretched, lost, overlapped, or duplicated, total area is preserved.
A dissection argument has two responsibilities. First, every piece of the original must appear in the new arrangement. Second, the pieces must exactly fill the new shape without overlap. If both conditions hold, area equality follows immediately from additivity of area.
Rectangle to Triangle
A rectangle of base b and height h has area bh. A triangle with the same base b would need height 2h to have the same area, because ½ × b × 2h = bh. This numerical relationship guides an equal-area construction.
Problem
A rectangle measures 10 cm by 4 cm. Construct dimensions for a triangle with the same 10 cm base and equal area.
- 1.Rectangle area = 10 × 4 = 40 cm².
- 2.Let the triangle base stay 10 cm.
- 3.Solve ½ × 10 × h = 40, giving h = 8 cm.
- 4.A triangle with base 10 cm and perpendicular height 8 cm has the same area.
A physical dissection can achieve the same transformation by cutting and repositioning portions of the rectangle. The exact cut may vary; the invariant is that the assembled triangle uses every original piece once.
Triangle to Rectangle
The reverse problem is equally useful. A triangle with base b and height h has area ½bh. A rectangle with the same base must therefore have height h/2. Alternatively, a rectangle with the same height must have base b/2.
Problem
A triangle has base 14 cm and height 9 cm. Give two different rectangles with the same area.
- 1.Triangle area = ½ × 14 × 9 = 63 cm².
- 2.Keep base 14 cm: rectangle height = 63 ÷ 14 = 4.5 cm.
- 3.Keep height 9 cm: rectangle base = 63 ÷ 9 = 7 cm.
- 4.Both 14 × 4.5 and 7 × 9 rectangles have area 63 cm².
Isosceles Triangles Make Symmetry Useful
In an isosceles triangle, an altitude from the top vertex to the base also bisects the base. This symmetry often makes dissection simpler. The two congruent right triangles can be moved and paired to form rectangular pieces.
Problem
An isosceles triangle has base 12 cm and height 8 cm. What rectangle with base 6 cm has the same area?
- 1.Triangle area = ½ × 12 × 8 = 48 cm².
- 2.A rectangle with base 6 cm must have height 48 ÷ 6 = 8 cm.
- 3.The altitude splits the triangle into two congruent right triangles that can be rearranged to suggest this 6 cm by 8 cm rectangle.
A Historical Geometric Connection
The Śulba-Sūtras are ancient Indian geometric texts associated with constructions in which prescribed shapes and areas mattered. Problems of transforming one region into another of equal area appear naturally in that setting. The mathematical idea is the same one used here: preserve area while changing shape.
This historical connection is valuable because it shows that area was not developed only as a collection of formulas. It was also a construction problem: how can a desired shape be built so that it contains exactly the required amount of surface?
Testing Whether an Area Transformation Is Valid
A picture that looks plausible is not enough. Ask: Were all original pieces used? Was any piece copied? Do any pieces overlap? Are there gaps? Were any lengths stretched? A valid dissection preserves the pieces themselves, so area preservation is guaranteed.
Cutting and moving pieces preserves area. Stretching a piece, scaling it, or changing its dimensions does not. A diagram should distinguish rigid movement from resizing.
Problem
A student cuts a 6 cm by 8 cm rectangle into three pieces, then enlarges one piece before assembling a triangle. Is the triangle guaranteed to have the same area?
- 1.No. Enlarging changes the area of that piece.
- 2.A valid equal-area dissection may translate, rotate, or reflect pieces, but it cannot resize them.
- 3.Area equality would need a separate calculation because the dissection rule was broken.
Quiz
What is the central reason a valid dissection preserves area?
A rectangle has base b and height h. A triangle with the same base and equal area needs height
A triangle has base b and height h. A rectangle with the same base and equal area needs height
Which action breaks an area-preserving dissection?
Why is an isosceles triangle often convenient for dissection?
Practice Problems
- A rectangle is 15 cm by 6 cm. Find the height of an equal-area triangle with the same 15 cm base.
- A triangle has base 20 cm and height 7 cm. Find a rectangle with the same base and equal area.
- Give a second rectangle with different dimensions but the same area as the triangle in the previous question.
- Explain the difference between rearranging a piece and scaling a piece.
- Design a paper-cut activity that transforms an isosceles triangle into a rectangle and explain why area is preserved.
- A rectangle has area 96 cm². Give three different triangle base-height pairs with the same area.
Key Takeaways
• Area belongs to the collection of pieces, so rigid rearrangement can change shape without changing total area. • A valid dissection uses every piece once, with no gaps or overlaps. • A rectangle bh matches a same-base triangle of height 2h. • A triangle ½bh matches a same-base rectangle of height h/2. • Symmetry can make dissections easier to construct and justify. • Equal-area transformation problems connect geometric formulas with construction-based reasoning.