Prism volume and cylinder measurement · Higher
| English | 中文 | Pinyin |
|---|---|---|
| cross-section/krɒs ˈsekʃn/ | 横截面 | héng jié miàn |
A tank has the same triangular end shape all along its length. Its volume depends on that end area and the distance the end shape extends.
- A tank has the same triangular end shape all along its length. Its volume depends on that end area and the distance the end shape extends.
- This lesson studies cross-section 横截面: A slice perpendicular to a prism’s length that remains constant along it.
Choose the mathematical structure
- For a right prism, volume is constant cross-sectional area times perpendicular length. Cuboid volume is lwh. A cylinder is a circular prism: V=πr²h. Total closed-cylinder surface area includes two circular ends and the curved surface 2πrh. Open containers omit the specified faces.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines cross-section?
A slice perpendicular to a prism’s length that remains constant along it.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
A triangular prism with end base 6 cm, end height 4 cm and length 10 cm has cross-section 12 cm² and volume 120 cm³. A cylinder of radius 3 cm and height 5 cm has volume 45π cm³. Its curved surface unwraps to a rectangle of width 2πr=6π and height 5, so curved area is 30π cm². Adding two ends gives total area 30π+18π=48π cm². An open-top tank of those dimensions has surface area 39π cm² because it keeps only one end. If a prism has volume 180 cm³ and cross-section 15 cm², its length is 12 cm. Convert 2000 cm³ to 2 litres, keeping volume conversion separate from surface area.
Prism volume and cylinder measurement
For a right prism, volume is constant cross-sectional area times perpendicular length
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find triangular-prism volume for end base 6, end height 4 and length 10.
(6×4/2)×10=120 cm³.
Test a tempting shortcut
- Use cross-sectional area, not perimeter, in the volume formula. Distinguish cylinder radius from diameter and identify whether end faces are present. A length times an area gives cubic units.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Prism volume equals cross-section perimeter times length. This claim is false. Explain which definition or assumption it violates.
Find the coefficient of pi in cylinder volume for r=3,h=5.
r²h=9×5=45.
Prism volume equals cross-section perimeter times length.
Use cross-sectional area, not perimeter, in the volume formula. Distinguish cylinder radius from diameter and identify whether end faces are present. A length times an area gives cubic units.
Interpret a new situation
- AQA G16/G17 includes cuboids, right prisms and cylinders. Draw the constant end shape, calculate it first and label the extrusion length. Preserve exact multiples of pi when asked.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the length of a prism with volume 180 and end area 15.
Length=volume/area=180/15.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A slice perpendicular to a prism’s length that remains constant along it. Choose the relationship, show the method, check its assumptions and interpret the result.