New Mexico Grade 8 Math

Different solids can have the same radius but different volume relationships. This original model compares a cylinder, a cone, and a sphere using their stated dimensions. Keep results in terms of π before making a comparison.

Try this original activity

A cylinder and a cone each have radius three units and height eight units. A sphere has radius three units. Find each volume in cubic units and order the solids from least to greatest volume. Explain why the cone and cylinder comparison uses both the same radius and the same height.

Work through the reasoning

The cylinder formula gives V = πr²h = π × 3² × 8 = 72π cubic units. The cone gives V = (1/3)πr²h = 24π cubic units. The sphere gives V = (4/3)πr³ = (4/3)π × 27 = 36π cubic units. Since π is positive, compare the coefficients: 24 < 36 < 72. The cone has the least volume, then the sphere, then the cylinder. The cone is one third of the cylinder because these particular solids share their radius and height; different dimensions would require a new calculation. A cylinder’s height is used in its formula, while the sphere’s radius determines its volume without a separate height variable. Keeping π exact avoids introducing rounding differences into the ordering.

Check the common mistake

The radius is three, not the diameter six. Use r² for the cylinder and cone and r³ for the sphere. The cone’s h is perpendicular height, not a slant length. State cubic units, since volume measures three-dimensional space. Equal volume does not imply that one shape fits inside another, and these abstract dimensions do not include container-wall thickness.

Try a related case

Keep the cylinder radius three but reduce its height to four. Its volume becomes 36π cubic units, equal to the sphere’s volume here. Explain why the sphere and cylinder still have different shapes. Then find the cone volume for radius three and height four: 12π, which is one third of that new cylinder.

Selected New Mexico mathematics references for this activity

The current New Mexico adopted content standards directory links the Common Core mathematics standards. The selected references below were checked in the state-hosted 2010 standards document. They identify this activity’s focus, rather than a complete course. Read the full requirements, mathematical practices, examples, and progression alongside the school’s curriculum.

Sources checked October 6, 2026. Read complete requirements, clarifications, limitations, and mathematical practices alongside the school’s sequence. This selected activity is an entry point for instruction.

Connect the explanation with practice

Use the lesson, worksheet, and quiz directory to choose a related skill. The existing resource links remain available; open each activity’s directions to understand its timer, tools, question count, and scoring.

The current state NM-MSSA resource page describes statewide mathematics and ELA assessment at the end of Grades 3–8, including Grade 8. Scores are one measure and should be considered alongside classroom and other assessment evidence. Confirm the learner’s assigned pathway and school directions. The assessment directory is updating resources for 2026–2027; this guide does not claim Spring2027 administration specifications from a Spring2026 manual. Independently published practice may use its own timer, tools, question count, and scoring; these are not official administration specifications or a prediction of a state performance level. Read the official NM-MSSA resource page.

Read New Mexico adopted content standards for later guidance. Keep one example of the student’s reasoning with the practice result. Choose a smaller model, a related question, or a new context according to the difficulty revealed in the worked activity.