Task Sets: Explain circle area and volume formulas - Common Core Math - HS Geometry

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Warm-Up

Which statement best describes an informal argument in mathematics?

A step-by-step deduction from axioms and definitions that proves a statement with certainty

Intuitive reasoning using visualization, approximation, or physical experiments that makes a result plausible

A guess that has not yet been verified by any evidence

A type of reasoning that is less important than formal proof and should be avoided

A regular polygon is inscribed in a circle. As the number of sides of the polygon increases, what happens to its perimeter?

The perimeter decreases and approaches zero

The perimeter stays the same regardless of the number of sides

The perimeter increases and gets closer to the circumference of the circle

The perimeter increases without bound, far exceeding the circumference

Which expression gives the area of a circle with radius $r$ ?

$2\pi r$

$\pi r^2$

$\pi r^3$

$\pi d$

Fluency Practice

$A regular hexagon inscribed in a circle of radius 1, with one side labeled 1$

A regular hexagon is inscribed in a circle of radius 1. Each side of the hexagon equals the radius, so each side has length 1. What is the perimeter of the hexagon, and how does it compare to the circumference $2\pi \approx 6.28$ ?

Perimeter = 6; less than $2\pi$ , confirming the polygon approximates but underestimates the circumference

Perimeter = 6; equal to $2\pi$ , confirming the formula is exact

Perimeter = $6\pi$ ; greater than $2\pi$ , so the polygon overestimates the circumference

Perimeter = $3.14 \times 6$ ; because $\pi = 3.14$ , the perimeter is about 18.85

In the wedge-dissection argument for circle area, a circle of radius $r$ is cut into many thin sectors and rearranged into a shape resembling a rectangle. Which dimensions does this near-rectangle have, and what area does it give?

Width $= 2\pi r$ , height $= r$ ; area $= 2\pi r^2$ — so the circle's area is $2\pi r^2$

Width $= \pi r$ , height $= r$ ; area $= \pi r^2$ — confirming $A = \pi r^2$

Width $= 2r$ , height $= \pi$ ; area $= 2\pi r$ — so the circle's area equals its circumference

Width $= r$ , height $= r$ ; area $= r^2$ — since $\pi$ cancels out in the rearrangement

A cylindrical pipe has radius $r = 3$ cm and height $h = 10$ cm. Using $V = \pi r^2 h$ , compute the volume. Express your answer in terms of $\pi$ (e.g., write $90\pi$ ).

$A square pyramid and a square prism with the same base B and height h, showing that three pyramids fill the prism$

A square pyramid and a square prism share the same square base (area $B$ ) and the same height $h$ . A student fills the pyramid with sand three times and pours it into the prism. The prism is exactly full. What does this demonstrate about the pyramid's volume?

$V_{\text{pyramid}} = Bh$ , the same as the prism, since both have the same base and height

$V_{\text{pyramid}} = \frac{1}{2}Bh$ , because the pyramid is half the size of the prism

$V_{\text{pyramid}} = \frac{1}{3}Bh$ , because three pyramids exactly fill one prism of the same base and height

$V_{\text{pyramid}} = \frac{1}{4}Bh$ , because the pyramid tapers to a point at the top

A square pyramid has a square base with side length 6 m and height $h = 8$ m. Compute its volume using $V = \frac{1}{3}Bh$ . The base area $B = 6^2 = 36$ m². Express your answer in cubic meters.

A cone has radius $r = 4$ cm and height $h = 9$ cm. Compute its volume using $V = \frac{1}{3}\pi r^2 h$ . Express your answer in terms of $\pi$ .

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Exercises: Informal Arguments for Volume and Area Formulas

Warm-Up

Fluency Practice