Spheres, cones, pyramids and frustums · Higher
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| frustum/ˈfrʌstəm/ | 截锥体 | jié zhuī tǐ |
A cone-shaped cup is cut flat at its top. The missing small cone must be removed from the original cone before the capacity is known.
- A cone-shaped cup is cut flat at its top. The missing small cone must be removed from the original cone before the capacity is known.
- This lesson studies frustum 截锥体: The remaining solid after a cone or pyramid is cut parallel to its base.
Choose the mathematical structure
- Pyramid and cone volumes are one third of base area times perpendicular height. A sphere has volume 4πr³/3 and area 4πr². A cone’s curved area is πrl using slant height l; its volume uses perpendicular height h. For a frustum subtract the removed similar solid. Composite surface area counts only exposed faces; joined faces are hidden.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines frustum?
The remaining solid after a cone or pyramid is cut parallel to its base.
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 cone with radius 3 cm and perpendicular height 4 cm has slant height 5 cm. Volume is 12π cm³, curved area 15π cm² and total closed area 24π cm². A sphere of radius 3 has volume 36π cm³ and area 36π cm², with different units. A square pyramid of base side 6 and height 4 has volume 6²×4/3=48 cm³; each triangular face has slant height √(4²+3²)=5, so lateral area is 4×(6×5/2)=60 cm² and total area 96 cm². A large cone r=6,h=8 loses a similar top cone r=3,h=4: frustum volume is 96π-12π=84π cm³. Its slant height is 10-5=5; curved area is 60π-15π=45π, and two circular ends add 36π+9π for total 90π cm².
Spheres, cones, pyramids and frustums
Pyramid and cone volumes are one third of base area times perpendicular height
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
Find the coefficient of pi in cone volume for r=3,h=4.
(1/3)×3²×4=12.
Test a tempting shortcut
- Perpendicular height and slant height are not interchangeable. A frustum is not a full cone of the leftover height. Shared composite faces do not contribute exposed area.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A cone’s slant height can always replace its perpendicular height in the volume formula. This claim is false. Explain which definition or assumption it violates.
Find square-pyramid volume for base side 6,height 4.
(1/3)×36×4=48.
A cone’s slant height can always replace its perpendicular height in the volume formula.
Perpendicular height and slant height are not interchangeable. A frustum is not a full cone of the leftover height. Shared composite faces do not contribute exposed area.
Interpret a new situation
- AQA G17 additional Foundation includes spheres, pyramids, cones, composite solids and frustums. Use similarity to find missing removed dimensions, then subtract volumes or exposed areas with matching units.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the coefficient of pi in frustum volume for large cone r=6,h=8 minus small r=3,h=4.
96-12=84.
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.
The remaining solid after a cone or pyramid is cut parallel to its base. Choose the relationship, show the method, check its assumptions and interpret the result.