Maryland Math G.GMD.A.1
The Standard
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
Maryland College and Career Ready Standards for Mathematics (2025) · EXPLAIN VOLUME FORMULAS AND USE THEM TO SOLVE PROBLEMS.
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students use polygons with more sides and thinner circle sectors to explain circumference and area formulas. They use stacked slices, three pyramids that fill a cube, and matching cross sections to explain cylinder, pyramid, and cone volumes.
What Mastery Looks Like
- A student can explain why a circle’s circumference is 2πr and its area is πr². They can justify cylinder volume as base area times height, and pyramid and cone volume as one-third of that amount.
Common Misconceptions
- Some students use πr² for the distance around a circle. Others leave out the one-third for pyramids and cones, or assume a cone and cylinder with the same base and height have equal volumes.
How to Assess It
- Exit ticket: A cone and a square pyramid each have base area 36 cm² and height 6 cm. Sketch a cube split into three pyramids, then use matching-height slices to explain why each volume is 72 cm³.
Lesson moves
Ways to Teach It
Cut a paper circle into 16 sectors and alternate them into a near rectangle; repeat with 32 sectors to see why area approaches πr².
Ask: If a cone and pyramid have equal base areas and heights, why do their slices at every matching height have equal areas?
Sketch three pyramids from one cube corner to its three opposite faces; ask pairs to explain why each holds one-third of the cube.
Wrap string around three round lids, measure each diameter, and compare circumference divided by diameter to estimate π before writing 2πr.
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Related Standards
- 7.GR.B.4.b
Solve problems involving volume of curved-surface solids, including cones, cylinders, and spheres—figures without edges.
- 8.GR.B.4.a
Use diagrams, examples, or informal reasoning to explain the Pythagorean Theorem and why it applies to all right triangles.
- 7.GR.B.3
Apply the formulas for the area and circumference of a circle to solve problems in context. Use informal reasoning and visual models to explain the relationship...
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