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3D Measurement & Cavalieri Lab

Compute surface area and volume for cylinders, cones, spheres, prisms, pyramids, and frustums. Slice solids with adjustable planes to discover cross-section shapes. Verify Cavalieri's Principle by comparing solids with equal cross-sectional areas at every height.

Guided Experiment: Volume Formulas Exploration

Which solid do you think has the largest volume for a given set of dimensions? How do you think doubling the radius affects volume compared to doubling the height?

Write your hypothesis in the Lab Report panel, then click Next.

Controls

Radius5.0
Height10.0

Results: Cylinder

Surface Area Formula
SA=2πr(r+h)SA = 2\pi r(r + h)
Volume Formula
V=πr2hV = \pi r^2 h
Surface Area Calculation
SA=2π(5)(5+10)=471.2389SA = 2\pi(5)(5 + 10) = 471.2389
Volume Calculation
V=π(5)2(10)=785.3982V = \pi(5)^2(10) = 785.3982
Surface Area
471.24
Volume
785.40
SA/V Ratio
0.6000

Solid Diagram

h=10r=5

Data Table

(0 rows)
#TrialSolidDimensionsSurface AreaVolumeSA/V Ratio
0 / 500
0 / 500
0 / 500

Reference Guide

Volume Formulas

Volume measures the space inside a 3D solid. Each shape has its own formula based on its geometry.

Vcyl=πr2h,Vcone=13πr2hV_{\text{cyl}} = \pi r^2 h, \quad V_{\text{cone}} = \frac{1}{3}\pi r^2 h
Vsphere=43πr3,Vprism=BhV_{\text{sphere}} = \frac{4}{3}\pi r^3, \quad V_{\text{prism}} = Bh

Notice the cone has exactly 1/3 the volume of a cylinder with the same base and height. The sphere formula depends only on its radius.

Surface Area Formulas

Surface area is the total area covering the outside of a solid. It includes all faces, bases, and curved surfaces.

SAcyl=2πr(r+h)SA_{\text{cyl}} = 2\pi r(r + h)
SAsphere=4πr2SA_{\text{sphere}} = 4\pi r^2

The surface-area-to-volume ratio (SA/V) is important in biology and engineering. Spheres have the lowest SA/V ratio of any shape.

Cross-Sections of Solids

A cross-section is the 2D shape you get when you slice a 3D solid with a flat plane.

  • Horizontal slice of a cylinder gives a circle
  • Horizontal slice of a cone gives a circle (smaller toward the apex)
  • Diagonal slice of a cylinder gives an ellipse
  • Vertical slice through a cone's axis gives a triangle

Understanding cross-sections is essential for calculating volumes using integration and for 3D printing (layer-by-layer construction).

Cavalieri's Principle

If two solids have the same height and their cross-sectional areas are equal at every level, then they have the same volume.

A1(h)=A2(h)  ∀ h  ⟹  V1=V2A_1(h) = A_2(h) \; \forall \, h \implies V_1 = V_2

This means a tilted (oblique) cylinder has the same volume as a right cylinder with the same base and height. Think of it like a stack of coins that stays the same total volume whether you push the stack sideways or keep it straight.