Measurement & Geometry Worksheet for 14-Year-Olds — LEGO SPIKE Prime | Australian Year 9 & AoPS

Instructions

You are designing a new delivery robot using the LEGO® SPIKE™ Prime set. Your robot needs a sturdy base, a flexible arm, and a container to hold its cargo. Use your knowledge of measurement and geometry to solve the following challenges and ensure your design is perfect!

Important Information:

• A "stud" is the small cylindrical bump on a LEGO brick. For these problems, assume 1 stud = 8 mm.
• Use π ≈ 3.14 where needed.
• Show your working where appropriate.

Part 1: The Robot's Base

The stability of your robot depends on a well-designed base.

1. The Support Brace: You use a right-angled triangular brace to support the main chassis. The two shorter sides of the brace are made from LEGO beams measuring 96 mm and 128 mm. What is the length of the longest beam (the hypotenuse) needed to complete the triangle?
   
   (Hint: Use Pythagoras' Theorem: a² + b² = c²)


2. The Platform Area: The main platform is a rectangular plate measuring 24 studs long and 16 studs wide. A circular hole with a radius of 2 studs is cut from the center to mount the motor. Calculate the total surface area of the top of the platform in square millimeters (mm²), excluding the hole.
   
   (Hint: Convert studs to mm first. Area of a rectangle = length × width. Area of a circle = πr²)

Part 2: The Robotic Arm

The arm needs to move precisely to pick up and drop off cargo.

3. Lifting Height: The robot's main arm segment is 150 mm long. When it lifts an object, the arm is positioned at a 40° angle to the horizontal platform. How high above the platform is the end of the arm? Round your answer to the nearest whole millimeter.
   
   (Hint: SOH CAH TOA. Which trigonometric ratio connects the angle, the hypotenuse, and the opposite side?)


4. Angle of Reach: To reach a high shelf, the arm must position its end-effector so it is 90 mm vertically above the arm's pivot point and 120 mm horizontally from the pivot point. What angle (θ) does the arm make with the horizontal at this position? Round your answer to one decimal place.

Part 3: The Cargo Holder

Your robot must be able to carry different containers.

5. Cylindrical Canister: The robot is designed to carry a cylindrical canister. The canister has a diameter of 64 mm and a height of 100 mm.
   
   a) What is the maximum volume of liquid it can hold in cubic millimeters (mm³)?
   (Volume of a cylinder = πr²h)
   
   b) What is this volume in cubic centimeters (cm³)?
   (Hint: 1 cm³ = 1000 mm³)


6. Composite Container: You build a custom cargo box shaped like a rectangular prism with a half-cylinder on top. The rectangular base is 80 mm long, 60 mm wide, and 50 mm high. The half-cylinder lies on top, sharing the 80 mm length and 60 mm width. What is the total volume of this container in mm³? Round your answer to the nearest whole number.

Part 4: The Challenge Problem

Think carefully about units and volume to solve this final task.

7. Brick by Brick: The SPIKE Prime Hub (the "brain" brick) has a volume of approximately 288 cm³. You want to build a solid rectangular tower with the same volume using only standard 2x4 stud LEGO bricks. A single 2x4 brick measures 16 mm × 32 mm × 9.6 mm. Approximately how many 2x4 bricks would you need to build a tower with a volume equal to or greater than the Hub's volume?
   
   (Hint: First, find the volume of one brick in cm³. Then determine how many bricks are needed to reach the target volume.)

Answer Key

1. Answer: 160 mm
   Working: Using Pythagoras' Theorem a² + b² = c².
   96² + 128² = c²
   9216 + 16384 = c²
   25600 = c²
   c = √25600 = 160 mm.


2. Answer: 23,895.04 mm²
   Working:
   Platform dimensions in mm: Length = 24 studs × 8 mm/stud = 192 mm. Width = 16 studs × 8 mm/stud = 128 mm.
   Hole radius in mm: 2 studs × 8 mm/stud = 16 mm.
   Area of rectangle = 192 mm × 128 mm = 24,576 mm².
   Area of hole = πr² = 3.14 × (16 mm)² = 3.14 × 256 = 803.84 mm².
   Total Surface Area = Area of rectangle - Area of hole = 24,576 - 803.84 = 23,772.16 mm².


3. Answer: 96 mm
   Working: The height is the side 'opposite' the angle, and the arm length is the 'hypotenuse'. We use sine.
   sin(θ) = Opposite / Hypotenuse
   sin(40°) = Height / 150 mm
   Height = 150 × sin(40°) ≈ 150 × 0.6428 ≈ 96.42 mm.
   Rounded to the nearest whole millimeter, the height is 96 mm.


4. Answer: 36.9°
   Working: The vertical distance is 'opposite' (90 mm) and the horizontal distance is 'adjacent' (120 mm). We use tangent.
   tan(θ) = Opposite / Adjacent
   tan(θ) = 90 / 120 = 0.75
   θ = tan⁻¹(0.75) ≈ 36.87°
   Rounded to one decimal place, the angle is 36.9°.


5. Answer: a) 321,536 mm³ b) 321.536 cm³
   Working:
   a) The diameter is 64 mm, so the radius (r) is 32 mm. Height (h) is 100 mm.
   Volume = πr²h = 3.14 × (32 mm)² × 100 mm = 3.14 × 1024 × 100 = 321,536 mm³.
   b) To convert mm³ to cm³, divide by 1000.
   321,536 mm³ / 1000 = 321.536 cm³.


6. Answer: 353,080 mm³
   Working:
   Volume of rectangular prism = length × width × height = 80 × 60 × 50 = 240,000 mm³.
   Volume of half-cylinder: The diameter is the width (60 mm), so the radius is 30 mm. The height (or length) is 80 mm.
   Volume = ½ × (πr²h) = 0.5 × 3.14 × (30)² × 80 = 0.5 × 3.14 × 900 × 80 = 113,040 mm³.
   Total Volume = 240,000 + 113,040 = 353,040 mm³.


7. Answer: 59 bricks
   Working:
   First, convert the brick dimensions to cm: 1.6 cm × 3.2 cm × 0.96 cm.
   Volume of one brick = 1.6 × 3.2 × 0.96 = 4.9152 cm³.
   Number of bricks needed = Total Volume / Volume of one brick
   Number = 288 cm³ / 4.9152 cm³ ≈ 58.59 bricks.
   Since you cannot use a fraction of a brick, you need to round up to the next whole number to meet or exceed the volume. You would need 59 bricks.