Understanding Archimedes' Method and Polygons

The video explains how Archimedes calculated π using the method of exhaustion. By inscribing polygons within a circle and increasing the number of sides, the perimeter of the polygon approaches the circle's circumference, thus approximating π. The video derives a formula for the perimeter of the inscribed polygon using trigonometry, specifically focusing on the sine of angle theta, which is determined by the number of polygon sides. As the number of sides increases, the approximation of π becomes more accurate.

What is the significance of the formula derived from Archimedes' method?

• It provides an exact value of π.

• It approximates the value of π.

• It determines the radius of a circle.

• It calculates the area of a circle.

What method did Archimedes use to calculate π?

• Method of integration

• Method of differentiation

• Method of substitution

• Method of exhaustion

What is the initial step in Archimedes' method of exhaustion?

• Creating a hexagon with a radius of 1/2

• Creating a triangle with a radius of 1/2

• Creating a circle with a radius of 1/2

• Creating a square with a radius of 1/2

What is the relationship between the radius and the circumference of the circle in Archimedes' method?

• The circumference is equal to 4π.

• The circumference is equal to π/2.

• The circumference is equal to 2π.

• The circumference is equal to π.

What happens when the number of sides of the inscribed polygon is increased?

• The polygon's area becomes zero.

• The polygon's perimeter decreases.

• The polygon becomes a square.

• The polygon resembles the circle more closely.

What is the relationship between the perimeter of the polygon and the circle's circumference?

• The perimeter is always greater than the circumference.

• The perimeter approximates the circumference.

• The perimeter is half of the circumference.

• The perimeter is unrelated to the circumference.

How is the length of a side of the polygon determined?

• By using the tangent of theta

• By using the sine of theta

• By using the cosine of theta

• By using the cotangent of theta

What is the formula for the perimeter of an equilateral polygon?

• n times cosine of theta

• n times cotangent of theta

• n times tangent of theta

• n times sine of theta

What is the role of trigonometry in deriving the perimeter formula?

• It is used to calculate the radius.

• It is used to calculate the area.

• It is used to determine the number of sides.

• It is used to find the length of a side.

What is the significance of point D in the construction of the polygon?

• Point D is the center of the circle.

• Point D is the midpoint of the circle.

• Point D bisects line BC.

• Point D is irrelevant to the construction.