New Hampshire Grade 8 Math

A right circular cylinder has equal circular cross-sections along its perpendicular height. Its volume is base area multiplied by height. The original model here isolates the cylinder part of a broader standard that also includes cones and spheres.

Try this original activity

An idealized right circular cylinder has radius 3 cm and height 8 cm. Find its volume exactly in terms of π, then approximately using π ≈ 3.14. Explain why multiplying circumference by height would not answer the volume question.

Work through the reasoning

The circular base area is πr² = π × 3² = 9π cm². Multiply this area by the perpendicular height: V = 9π × 8 = 72π cm³. With π ≈ 3.14, the approximate volume is 226.08 cm³. The units provide a check: square centimeters times centimeters gives cubic centimeters. Circumference times height would instead give a square-unit result associated with the cylinder’s curved surface, not its interior volume. The radius is three centimeters; the diameter would be six. The diagram is drawn in perspective and its top appears oval on the screen, but the mathematical solid has a circular base.

Check the common mistake

Using six as the radius would square a value twice as large and produce four times the correct volume. Multiplying 3 × 8 leaves out both π and the squared radius. For a real container, outer dimensions do not necessarily give inner capacity because wall thickness takes space. This task supplies dimensions of an idealized solid and makes no claim about a real container’s usable volume.

Try a related case

Keep the radius at 3 cm and halve the height to 4 cm. The base area stays 9π cm² and the volume halves to 36π cm³. If the height instead stays 8 cm while the radius doubles to 6 cm, volume becomes 288π cm³, four times the original. Explain the different changes from the formula before calculating.

Selected Common Core references for this activity

New Hampshire’s current school-approval rules, Ed 306.34, allow local boards to use the adopted 2010 Common Core mathematics standards or approved alternative academic standards. Confirm the school’s standards choice and local competencies first. The selected Common Core references below were checked in the original publisher’s 2010 document. They identify this activity’s focus, rather than a complete local course. Read the full requirements, mathematical practices, examples, and progression with 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 2025–2026 public NH SAS manual describes general mathematics testing in Grades 3–8, including Grade 8, using a computer-adaptive assessment. Confirm the learner’s assigned pathway, tools, and the school’s current administration directions. The reviewed manual is the 2025–2026 edition; confirm later-year directions with the school. 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 public NH SAS manual resource page.

Read New Hampshire local curriculum rules 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.