Wheel Wonders Slides Circle Carnival
The Geometry of Circular Motion
Middle School Math • Design Project
Anatomy of the Wheel
1
Radius (\(r\))
Center to the edge. Determines the size.
2
Diameter (\(d\))
Edge to edge through center (\(d = 2r\)).
3
Circumference (\(C\))
The total "track" length (\(C = 2\pi r\)).
Radius
Spacing the Seats
To space seats perfectly, we divide the full 360° rotation by the number of seats (\(n\)).
The Formula
\(\frac{360^\circ}{n}\)
\(n = \text{number of seats}\)
30°
Arc Length: Physical Distance
Arc length is the actual distance a passenger travels along the curve from one point to another.
The Logic
Take the fraction of the circle and multiply it by the total Circumference .
\(\text{Arc} = \frac{\theta}{360} \cdot C\)
Pro Engineer Tip
If you have 12 seats, each "gap" is exactly \(1/12\) of the total distance around the wheel. No complex angles needed if you know your seat count!
Ferris Wheel Physics Worksheet Ferris Wheel Physics
Guided Practice: Circle Geometry & Central Angles
Name: ______________________
Date: _______________________
Circumference
\(2\pi r\)
Diameter
\(2r\)
Central Angle
\(360^\circ / n\)
Arc Length
\(\frac{\theta}{360} \cdot C\)
Part 1: Basic Calculations
1. Circumference Check
A wheel has a radius of 45 feet. Find the total distance of one rotation. (Use \(\pi \approx 3.14\))
2. Radius Check
If a wheel's diameter is 120 feet, what is the radius length?
3. Seat Spacing
A wheel has 16 gondolas spaced equally. Calculate the central angle between each seat.
4. Working Backwards
A wheel with a central angle of \(45^\circ\) between seats has how many total seats?
Part 2: Complex Scenarios
Data Set: Radius = 30 feet | Number of Seats = 12
5. Calculate Circumference
Find the total circumference for this specific wheel.
6. Calculate Interval Angle
Find the central angle for this 12-seat wheel.
7. Physical Distance (Arc Length)
What is the distance along the rim between two adjacent seats?
Master Engineer Challenge
You are at Seat 1. Your friend is at Seat 4. If the radius is 50ft and there are 12 seats total, find the total arc distance between you. (Hint: How many 'gaps' are between Seat 1 and Seat 4?)
Designer Blueprint Guide Designer Blueprint Guide
Ferris Wheel Engineering Project
Project Scope
Design a custom Ferris wheel attraction. You must calculate the geometry and provide a scaled blueprint. Ensure all calculations are shown clearly. Refer to the Project Grading Rubric for evaluation standards.
Engineering Constraints
Max Radius: 40 feet
Seat Options: 8, 10, or 12
Drawing Scale: 1 in = 5 ft
01 Design Specifications
Attraction Name
Design Theme
Target Radius (\(r\))
Total Seat Count (\(n\))
02 Technical Geometry Calculations
A. Full Circumference (\(C\))
\(2\pi r\)
B. Interval Angle (\(\theta\))
\(360 / n\)
C. Arc Length (1 Seat to Next)
\(\frac{\theta}{360} \cdot C\)
D. Arc Length (Seat 1 to 4)
3 intervals
03 Technical Blueprint Drafting
Draft your design using a compass and protractor. Use your central angle calculation to place seats precisely.Scale: 1 inch = 5 feet. (A 40ft radius is drawn as an 8-inch diameter circle).
Drafting Canvas
Project Grading Rubric Project Grading Rubric
Design Standards & Evaluation Metrics
Category Exceptional (10) Proficient (8) Developing (6) Beginning (4) Calculations 100% accurate calculations with all work and units shown. Mostly accurate. Minor rounding or 1 calculation error. Multiple math errors. Units are missing or inconsistent. Math is missing or consistently incorrect. Angle Precision Seat placement perfectly matches calculated angles. Placement is close with only minor deviations. Angles on drawing do not match calculations. No evidence of angle measurement used. Drafting Professional quality. Accurate scale, labels, and theme. Neat drawing with scale and most labels included. Drawing lacks neatness or missing critical labels. Draft is incomplete, messy, or lacks a scale. Concepts Mastery of radius/angle/arc relationship is clear. Correct application of circle geometry formulas. Some confusion between core circle concepts. No understanding of circle geometry evident.
Engineer Review Notes
Project Score
/ 40
Math: ___ / 20
Draft: ___ / 20
Circle Carnival Answer Key Teacher Solution Key
Circle Carnival Project Reference
Worksheet Solutions
1. SKY SCREAMER (\(r=45\))
\(C = 2 \cdot 3.14 \cdot 45 = \mathbf{282.6\text{ ft}}\)
2. DIAMETER TO RADIUS
\(r = 120 / 2 = \mathbf{60\text{ ft}}\)
3 & 4. SEATS & ANGLES
3. \(360 / 16 = \mathbf{22.5^\circ}\)
4. \(360 / 45 = \mathbf{8\text{ seats}}\)
PART 2: DATA SET (\(r=30, n=12\))
5. \(C = \mathbf{188.4\text{ ft}}\)
6. Angle \(= \mathbf{30^\circ}\)
7. Arc Dist \(= \mathbf{15.7\text{ ft}}\)
Master Engineer Challenge
Calculating Seat 1 to Seat 4 Distance:
12 seats means \(\theta = 30^\circ\).
Total Angle = \(3 \times 30 = \mathbf{90^\circ}\).
Arc Distance = \(\frac{90}{360} \cdot (314) = \mathbf{78.5\text{ ft}}\).
Project Sample Data
10-seat design | 50ft radius
<table class="w-full text-[12px] border-collapse"><tbody><tr class="bg-emerald-50"><td class="p-3 border-b border-emerald-100 font-bold">Circumference</td><td class="p-3 border-b border-emerald-100 text-right">314 ft</td></tr><tr><td class="p-3 border-b border-emerald-100 font-bold">Central Angle</td><td class="p-3 border-b border-emerald-100 text-right">36°</td></tr><tr class="bg-emerald-50"><td class="p-3 border-b border-emerald-100 font-bold">Arc length (1-2)</td><td class="p-3 border-b border-emerald-100 text-right">31.4 ft</td></tr><tr><td class="p-3 font-bold">Arc Length (1-4)</td><td class="p-3 text-right font-bold text-emerald-700">94.2 ft</td></tr></tbody></table>
Teacher Note
The most common error is Seat 1 to Seat 4 being calculated as 4 intervals. Remind students to count the spaces between gondolas.
