Lemon Calculus

Source: Wikimedia

Type: Investigation

Theme: Calculus

Grades: 5, 6, 7, 8, 9, 10

Source: Youcubed.org

Learning Target: Students will calculate the volume of a lemon using various techniques: estimation, displacement, and Riemann sums.



  1. One lemon per group of two
  2. Centimeter ruler
  3. Graduated cylinder or transparent 500 mL measuring cup with measuring lines
  4. Knife
  5. Cutting surface
  6. Sugar and container (if making lemonade)

What to do

  1. Divide into groups.
  2. Discussion - Before students are given tools for measuring the volume of the lemon, students should discuss their ideas about how to find volume of a lemon. The teacher should remind younger students that volume of a cube is length x width x height. Groups should capture this brainstorming session on paper or digitally as an artifact for submission.
  3. Groups will estimate their volume using a centimeter ruler.
  4. Initially, the teacher will use the displacement method to find the volume of a lemon using a graduated measuring cup. Teacher will compare cubic centimeters with milliliters.
  5. The teacher will explain the technique of Riemann sums for finding area under a curve. Students will be using a similar method for finding the volume of a lemon. 
  6. The teacher will demonstrate how to slice the lemon to determine volume by summing the estimated volume of each cylindrical slice (π x radius squared x height): https://www.desmos.com/calculator/rwstvakjfd
  7. Students should compare their three values: (a) estimation, (b) displacement, (c) Riemann sum.

What to Submit

  1. One Sway or PowerPoint 
    1. Photographic documentation of the process 
    2. One paragraph explanation (copy/capture your brainstorming sessions)
    3. Volume of lemon using estimation (cubic centimeters)
    4. Volume of lemon using water displacement (mL)
    5. Volume of lemon in cubic centimeters using 'calculus'


(This task was inspired by an activity used by Laura Evans and Carlos Cabana while at San Lorenzo High School, California)

Exit Ticket
CCSS Math Practice
  • I can reason abstractly and quantitatively.
  • I can attend to precision.
NGSS Crosscutting Concepts
  • Scale, Proportion, and Quantity