Illinois Grade 8 Math

Volume formulas describe three-dimensional space, so the chosen dimensions and units matter as much as substitution. A cylinder and cone with matching circular bases and perpendicular heights give a useful comparison. The activity uses ideal geometric solids rather than actual container capacities.

Try the worked activity

A model cylinder and a model cone each have radius 2 cm and perpendicular height 5 cm. Find their volumes exactly in terms of π. Compare the results, then explain what would go wrong if a 4 cm diameter were substituted as the radius.

Follow the reasoning

The circular base area is πr² = π × 2² = 4π square centimeters. For the cylinder, V = πr²h gives 4π × 5 = 20π cubic centimeters. For the cone, V = (1/3)πr²h gives 20π/3 cubic centimeters. The cylinder therefore has three times the cone’s volume for these matching base and height dimensions. Using π ≈ 3.14 gives approximately 62.8 cm³ and 20.93 cm³; the expressions with π remain the exact values.

Check the explanation

The radius is half the diameter. Substituting four as the radius would multiply the base area and volume by four because the radius is squared. The cone’s height is the perpendicular distance from its base plane to its vertex, not a sloping side length. Also distinguish square centimeters for base area from cubic centimeters for volume. Matching height alone does not establish the three-to-one comparison: the two circular bases must also match. A real container’s wall thickness and usable interior dimensions would require additional information.

Use the next session

1. Identify the solid and distinguish radius from diameter.
2. Use the formula with the appropriate perpendicular height if required.
3. Check the dimension, exact expression, approximation, and cubic unit.

Compare a cylinder and cone with radius 3 cm and height 5 cm: their exact volumes are 45π cm³ and 15π cm³. Then apply the sphere formula V = (4/3)πr³ to a sphere of radius 3 cm, obtaining 36π cm³. Explain why the sphere calculation does not use the cylinder’s five-centimeter height. Change one dimension at a time to study how it affects the result, and use a labeled sketch to keep the dimensions clear. These examples cover formula use for selected models; wider geometric problems require identifying the correct solid and the available measurements.

Selected references from Illinois’s adopted mathematics document

The Illinois mathematics standards directory links the Illinois Learning Standards document incorporating Common Core. These selected references identify the worked activity’s focus. The expectations include numbered requirements, examples, and mathematical practices. Read the full document and confirm the school’s current instructional sequence.

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.

Grade 8 is within Illinois’s IAR mathematics testing range. The linked learning resources have their own directions and scoring; a practice score does not establish an official state performance level. Read Illinois’s assessment overview.

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.