Final answer:
To construct a regular hexagon, use a compass to draw arcs from six evenly spaced points on a circle's circumference and connect them with a straightedge. For an equilateral triangle, draw a circle's diameter, create arcs from both ends, and connect the points. Use circumference and area formulas to check your work.
Step-by-step explanation:
Constructing a Regular Hexagon and an Equilateral Triangle Within a Circle
To construct a regular hexagon within a circle using geometry tools, you’ll need a compass and a straightedge. Begin by drawing a circle of any radius. Place the compass at any point on the circle’s circumference and draw an arc across the circle. Without changing the compass width, repeat this process from the new intersection point. Continue around the circle until you have six arc intersections on the circumference. Connect these points with a straightedge to form the hexagon.
For constructing an equilateral triangle inscribed within a circle, draw a circle and with the compass set to the radius of this circle, make an arc from anywhere on the circumference. Make another arc the same way from the new intersection, creating two points. Draw a straight line through these two points across the circle; this line is the diameter. Now, without changing the compass width, place the compass on one end of the diameter and draw an arc within the circle. Do the same from the other end of the diameter. The two arcs will intersect above the diameter line, forming an equilateral triangle with the diameter points.
Remember, the perimeter of a circle is circumference (2πr) and the area is πr². These measurements can help you in verifying the accuracy of the constructed shapes with relation to the enclosing circle.