The Eternal’s Fullness
Edition 15, Lesson 15.3: The Area of a Circle
Part I: The Filling of the Center
Use the formula $A = \pi r^2$. Use 3.14 for $\pi$. Round to the nearest tenth.
1.
A circular garden has a **Radius** of 4 meters. What is its Area?
Area = ________ $m^2$
2.
A communal shield has a **Radius** of 12 inches. How much surface area must be polished?
Area = ________ $in^2$
3.
The base of a round pillar has a **Radius** of 2 cubits. What is its footprint area?
Area = ________ $cubits^2$
The Lawyer's Check:
Did you square the radius FIRST? Remember: $4^2 = 16$. Multiply by Pi last! If you are given the Diameter, find the radius root before you fill the space.
Part II: The Bridge to the Field (Diameter to Area)
Divide the diameter by 2 first, then calculate the area.
4.
A circular rug has a **Diameter** of 6 feet. What is its area?
Area = ________ $ft^2$
5.
A round pond is 20 meters across (**Diameter**). What is the area of the water's surface?
Area = ________ $m^2$
Part III: The Sector Challenge (Portions of the Fullness)
A Sector is a "piece of the pie." Calculate the area of the portion.
6.
A circular cake has a **Radius** of 10 cm. If the cake is cut into 4 equal slices, what is the area of one slice?
Sector Area = ________ $cm^2$
7.
A sanctuary window is a semi-circle (half a circle) with a **Diameter** of 6 feet. What is the area of the glass?
Semi-circle Area = ________ $ft^2$
Part IV: Concentric Rings (Growth Layers)
To find the area of a ring, subtract the small circle from the large circle.
8.
The Outer Courtyard:
A circular building has a radius of 5 meters. It is surrounded by a grass ring that extends out to a radius of 10 meters.
1. Calculate the Area of the large circle ($r=10$).
2. Calculate the Area of the building ($r=5$).
3. Subtract to find the Area of the grass ring.
Grass Area = ________ $m^2$
Part V: The Inverse Fullness (Area to Radius)
If you know the Area, can you find the Radius root? $r = \sqrt{A / \pi}$.
9.
A circle has an Area of 314 $cm^2$. What is its radius?
Radius = ________ cm
10.
The sanctuary courtyard has a circular area of 1256 $cubits^2$. How far is it from the center to the edge?
Radius = ________ cubits
Part VI: Kingdom Modeling
11.
The Altar of Gold:
A circular altar has a diameter of 4 feet. Gold leaf costs 5 coins per square foot.
1. Find the area of the altar top.
2. Calculate the total cost to cover the altar.
Total Cost: ________ coins
12.
The Spreading Gospel:
A ripple of grace spreads in a circle. In the first year, it has a radius of 10 miles. In the second year, the radius doubles to 20 miles.
1. Calculate Area 1.
2. Calculate Area 2.
3. How many times larger is the second year's harvest than the first?
Part VII: The Echad Extension (Transmission)
13.
The Mosaic Challenge:
Use a set of identical square objects (like coins or tiles) to try and "fill" a circular shape (like a dinner plate). How many squares fit entirely inside? How many squares touch the edge and have "empty space" around them? Explain to someone why the formula $\pi r^2$ is more "Faithful" than just counting squares.
Part VIII: Logic and Reflection
14.
The Law of the Square:
Explain why doubling the Radius ($r \times 2$) leads to quadrupling the Area ($A \times 4$). Use the formula $\pi r^2$ to prove your answer. What does this tell us about the power of focus?
15.
Fullness vs. Boundary:
In your own words, why is "Area" a better measure of a person's "Fullness" than "Circumference"? How does the area reflect the internal state of the heart? Why is God interested in the filling and not just the fence?
Part IX: The Third Dimension (A Preview of Volume)
If you stack circles, you get a Cylinder. Volume = Area of base x Height.
16.
A circular cistern has a radius of 3 meters and a height of 5 meters.
1. Calculate the Area of the base ($\pi r^2$).
2. Multiply by the height to find the total Volume.
Volume = ________ $m^3$
"I vow to fill my life with the presence of the Father. I will not be content with a mere line, but will seek the squared abundance of a heart rooted in the Center. I will be a faithful steward of the fullness He has given me."
[VOLUME 2 WORKBOOK SPEC: 15.3]
This workbook applies the area formula to various contexts. It reinforces the importance of the squaring operation and the conversion between diameter and radius.
Total Practice Items: 11
Theological Anchor: Abundance/Fullness