Prism volume and cylinder measurement · Higher
مُقدَّم| English | المعنى |
|---|---|
| cross-section/krɒs ˈsekʃn/ | المقطع العرضي |
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.
- This lesson studies cross-section 横截面: A slice perpendicular to a prism’s length that remains constant along it.
The method
- 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.
Why does a prism’s volume use cross-sectional AREA times length rather than perimeter times length?
الصعوبة 1/5 · 100 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
Each slice of the prism contributes its AREA of material stacked through the length. A perimeter is a boundary length - multiplying it by length would give a surface-like quantity, not a volume.
A worked case
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.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
(6×4/2)×10=120 cm³.
Where it goes wrong
- 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.
Not true: Prism volume equals cross-section perimeter times length. Volume equals cross-sectional area times length; perimeter times length gives the area of the side faces.
Find the coefficient of pi in cylinder volume for r=3,h=5.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
r²h=9×5=45.
Prism volume equals cross-section perimeter times length.
الصعوبة 1/5 · 100 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
Volume equals cross-sectional area times length; perimeter times length gives the area of the side faces.
Applying it
- 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.
Find the length of a prism with volume 180 and end area 15.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
Length=volume/area=180/15.
Match each prism quantity to its formula.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
The curved surface unwraps to a rectangle of width 2πr; closed ends add two circles, and an open tank omits one.
In your course
- AQA GCSE Mathematics (8300), Higher tier, specification section 3.4.
- Paper 1 is non-calculator: keep fractions, surds and π exact and show each step. On calculator papers, still write the expression before its value.
- Cross-section: A slice perpendicular to a prism’s length that remains constant along it.
- For a right prism, volume is constant cross-sectional area times perpendicular length.
- Watch for: use cross-sectional area, not perimeter, in the volume formula.