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Cavalieri's principle in 2D

If two figures have the same height and the same width at every point along that height, they have the same area.

Cavalieri's principle in 2D

Key idea: If two figures have the same start color #7854ab, start text, h, e, i, g, h, t, end text, end color #7854ab and the same start color #a75a05, start text, w, i, d, t, h, end text, end color #a75a05 at every point along that height, they have the same start color #208170, start text, a, r, e, a, end text, end color #208170.
Wait a second, you already know this! Check it out.
Rectangle with a base of 4 units and height of 6 units.
What is the area of the rectangle?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Parallelogram with a base of 4 units and height of 6 units.
What is the area of the parallelogram?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

According to Cavalieri's principle, these figures have the same area because they have the same height left parenthesis, 6, right parenthesis and the same width left parenthesis, 4, right parenthesis as each other at every point along that height.

Cavalieri's principle with varied widths

Sometimes a figure has different widths at different heights. Cavalieri's principle still works.
According to Cavalieri's principle, which of the following figures must have the same area?
Choose all answers that apply:
Choose all answers that apply:

Try this

What is the area of the following figure?
  • Your answer should be
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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