Question

practice 6 (from unit 4, lesson 12)
a regular hexagon and a regular octagon are both inscribed in the same circle. which of these statements is true?
a the perimeter of the hexagon is less than the perimeter of the octagon, and each perimeter is less than the circumference of the circle.
b the perimeter of the octagon is less than the perimeter of the hexagon, and each perimeter is less than the circumference of the circle.
c the perimeter of the hexagon is greater than the perimeter of the octagon, and each perimeter is greater than the circumference of the circle.
d the perimeter of the octagon is greater than the perimeter of the hexagon, and each perimeter is greater than the circumference of the circle.

Explanation:

1. For a regular polygon inscribed in a circle, the more sides the polygon has, the closer its perimeter is to the circle's circumference, and the longer its perimeter (because each side length is related to the chord length, and more sides with appropriate chord lengths sum to a longer perimeter as the number of sides increases, approaching the circumference).
2. A regular hexagon has 6 sides, a regular octagon has 8 sides. So the octagon has more sides, so its perimeter is longer than the hexagon's.
3. Also, the perimeter of any regular polygon inscribed in a circle is less than the circumference of the circle (since the polygon is inside the circle, and the circumference is the length around the circle, while the polygon's perimeter is a polygonal path inside, with each side being a chord, and the sum of chords is less than the arc length around the circle, and summing all arcs gives the circumference). So the hexagon's perimeter < octagon's perimeter < circumference of the circle. Wait, no—wait, as the number of sides increases, the perimeter approaches the circumference. So a hexagon (6 sides) has a perimeter less than an octagon (8 sides), and both are less than the circumference? Wait, no—wait, when you inscribe a polygon in a circle, the perimeter of the polygon is the sum of the chord lengths. The length of a chord is less than the length of the corresponding arc (since the chord is a straight line, the arc is curved, and the shortest path between two points on a circle is the chord, but the arc length is longer? Wait, no—wait, the chord length is \( 2r\sin(\frac{\pi}{n}) \) for a regular \( n \)-gon inscribed in a circle of radius \( r \), and the arc length is \( r\theta \) where \( \theta=\frac{2\pi}{n} \), so chord length is \( 2r\sin(\frac{\pi}{n}) \), arc length is \( \frac{2\pi r}{n} \). Let's compare chord length and arc length: \( \sin(x) < x \) for \( x>0 \), so \( 2r\sin(\frac{\pi}{n}) < 2r \cdot \frac{\pi}{n} \), so chord length < arc length. Then the perimeter of the polygon is \( n \times \) chord length \( = 2nr\sin(\frac{\pi}{n}) \), and the circumference is \( 2\pi r \). Let's check for \( n = 6 \): perimeter \( = 6 \times 2r\sin(\frac{\pi}{6}) = 6 \times 2r \times 0.5 = 6r \). Circumference is \( 2\pi r \approx 6.28r \). So hexagon perimeter \( 6r < 6.28r \) (circumference). For \( n = 8 \): perimeter \( = 8 \times 2r\sin(\frac{\pi}{8}) \approx 8 \times 2r \times 0.3827 \approx 6.123r \). Wait, that's more than 6r (hexagon's 6r). So octagon perimeter (≈6.123r) is more than hexagon's (6r), and both are less than circumference (≈6.28r). So the correct statement is that the perimeter of the hexagon is less than the perimeter of the octagon, and each perimeter is less than the circumference of the circle. So option A.

Answer:

A. The perimeter of the hexagon is less than the perimeter of the octagon, and each perimeter is less than the circumference of the circle.