Spheres, cones, pyramids and frustums · Higher
הוצג| English | משמעות |
|---|---|
| frustum/ˈfrʌstəm/ | פסוגה |
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.
- This lesson studies frustum 截锥体: The remaining solid after a cone or pyramid is cut parallel to its base.
The method
- 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.
A cone has radius 3 and perpendicular height 4. Why is its slant height 5 NOT used in V=πr²h/3?
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
Volume depends on the perpendicular height - the direction in which cross-sectional areas accumulate. The slant height 5 (from the 3-4-5 triangle) belongs to the curved-surface formula πrl instead.
A worked case
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.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
(1/3)×3²×4=12.
Where it goes wrong
- 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.
Not true: A cone’s slant height can replace its perpendicular height in the volume formula. Cone volume uses the perpendicular height; the slant height is longer and gives too large a volume.
Find square-pyramid volume for base side 6,height 4.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
(1/3)×36×4=48.
Cone volume uses the perpendicular height.
רמת קושי 1/5 · 100 מטבעות + בונוס קטן בהשלמה הראשונה היום
True. Cone volume uses the perpendicular height; the slant height is longer and gives too large a volume.
Applying it
- 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.
Find the coefficient of pi in frustum volume for large cone r=6,h=8 minus small r=3,h=4.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
96-12=84.
Match each solid to its volume formula.
רמת קושי 2/5 · 140 מטבעות + בונוס קטן בהשלמה הראשונה היום
The one-third factor comes from tapering to a point; a frustum keeps the difference of two similar cones, not a cone of the leftover height.
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.
- Frustum: The remaining solid after a cone or pyramid is cut parallel to its base.
- Pyramid and cone volumes are one third of base area times perpendicular height.
- Watch for: perpendicular height and slant height are not interchangeable.