Circular Pastures and Arbelos
In this activity, you will explore the proportional relationship between radius (or diameter) and circumference in circles as well as the proportional relationship between the square of radius and area. You will formulate conjectures using dynamic geometry software.
COMMON CORE STATE STANDARD – HS GEOMETRY - MATHEMATICAL PRACTICE
Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures.
MEASURING PASTURES USING DYNAMIC SOFTWARE
Observe Pastures A, B, C and D. Use the "Distance or Length" tool to measure the arcs' length, from there, find the perimeter of each pasture. Then, calculate the area of each pastures.
PASTURE A
Pasture A's perimeter is:
Pasture A's area is:
PASTURE B
Pasture B's perimeter is:
Pasture B's area is:
PASTURE C
Pasture C's perimeter is:
Pasture C's area is:
PASTURE D
Pasture D's perimeter is:
Pasture D's area is:
Make a conjecture about the relationship among the perimeter and area of Pastures A, B, C, and D.
EXPLAINING THE CONJECTURE
In the preceding section, you formulated the following conjecture:
All four Pastures have the same perimeter but not the same area. Pasture B has the largest area and Pasture C has the least area.
Use proportionality and distributivity properties to verify that the conjecture is true for any set of diameters d1, d2, d3, ..., dn; where d1 + d2 + d3 + ... + dn = diameter of the big circle.
FURTHER EXPLORATION - ARBELOS
The figures representing the pastures in the previous section are related to a classic figure, studied since the time of Archimedes, called the arbelos – the region bounded by three semicircles that are tangent in pairs with diameters that lie on the same line (i. e. the red region in the figure)
Move point P to observe different arbelos.
The below figure shows how to construct an arbelos.
Use the "Area" tool to find the area of the above arbelos.
Construct a circle with diameter PR and calculate its area.
Make a conjecture about the areas:
Proof your conjecture by building a logical progression of statements.