Volume of Frustums and Spheres

By the end of this lesson, learners should be able to

1. Deduce and apply the subtraction method to determine the volume of conical and pyramidal frustums.
2. Calculate the volume of a sphere using the formula $V=\frac{4}{3}\pi r^3$.
3. Solve practical multi-step problems involving frustums, spheres, and composite solids in real-life contexts.

Key vocabulary

Frustum

Part of a cone or pyramid left after cutting off its apex end.

We subtract the small cut-off cone volume from the full cone volume.

Truncated Solid

A three-dimensional solid shape cut short by a plane section parallel to its base.

Truncated solids like bucket frustums are evaluated by subtracting similar apex volumes.

Linear Ratio

The ratio comparing corresponding straight length dimensions of two similar geometric figures.

The linear ratio of eight to twenty gives the top height as twelve centimetres.

Sphere

A round three-dimensional geometric solid where every surface point is equidistant from the centre.

The volume of the leather sphere is four thousand eight hundred fifty-one cubic centimetres.

Hemisphere

Half of a sphere bounded by a circular plane interface.

The storage tank volume combines a frustum volume and a hemisphere volume.

Lesson preview

Slide 1 · Understanding Frustum Volume by SubtractionOpening · 10 min

A frustum is formed when the apex of a cone or pyramid is sliced off parallel to its base. Visualising the original complete shape allows us to find its volume by subtraction. The volume of the frustum equals the volume of the large initial solid minus the volume of the small solid that was cut away. Because the small solid is similar to the original whole solid, linear dimensions are proportional.

• A frustum is formed by slicing a cone or pyramid parallel to its base and removing the part containing the apex.
• The volume of a frustum equals the volume of the original solid minus the smaller cut-off solid.
• The height and base dimensions of the small cut-off solid are proportional to those of the whole solid.
• For a conical frustum, subtract the smaller cone volume from the larger original cone volume.

Worked example. Identify heights: the complete cone (apex below the bucket, up to the top rim) has height $H=20\text{ cm}$; the small cone cut off below the bottom has height $h=20-10=10\text{ cm}$. Calculate volume of the complete cone with top radius $R=12\text{ cm}$: $V_1=\frac{1}{3}\pi R^{2}H=\frac{1}{3}\times 3.142\times {12}^2\times 20=3016.32{\text{ cm}}^3$. Calculate volume of the small cone below the bottom, with radius $r=6\text{ cm}$: $V_2=\frac{1}{3}\pi r^{2}h=\frac{1}{3}\times 3.142\times 6^2\times 10=377.04{\text{ cm}}^3$. Subtract the small cone volume from the complete cone volume: $V_{\text{frustum}}=V_1-V_2=3016.32-377.04=2639.28{\text{ cm}}^3$. Answer: $2639.28{\text{ cm}}^3$.

Check question: A rectangular-based pyramid frustum has bottom base dimensions $15\text{ cm}\times 10\text{ cm}$, top base dimensions $6\text{ cm}\times 4\text{ cm}$, total original pyramid height $20\text{ cm}$, and top pyramid height $8\text{ cm}$. Calculate its volume.

1. $936{\text{ cm}}^3$
2. $1000{\text{ cm}}^3$
3. $1064{\text{ cm}}^3$
4. $2808{\text{ cm}}^3$

Show answer

$936{\text{ cm}}^3$. Volume of full pyramid = $\frac{1}{3}\times 15\times 10\times 20=1000{\text{ cm}}^3$. Volume of top pyramid = $\frac{1}{3}\times 6\times 4\times 8=64{\text{ cm}}^3$. Frustum volume = $1000-64=936{\text{ cm}}^3$.

Slide 2 · Volume of Pyramid FrustumsModel · 10 min

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• All 4 slides with diagrams and class activities
• Teacher notes, SNE adaptations and assessment checks
• Homework and the other 8 lessons in this sub-strand

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All 9 lessons in Surface Area and Volume of Solids

1. 1Surface Area of Prisms and PyramidsDetermine the surface area of prisms, pyramids, cones, frustums and spheres
2. 2Surface Area of Pyramids, Cones, and FrustumsDetermine the surface area of prisms, pyramids, cones, frustums and spheres
3. 3Surface Area of Frustums, Spheres, and HemispheresDetermine the surface area of prisms, pyramids, cones, frustums and spheres
4. 4Surface Area of Composite SolidsCalculate the surface area of composite solids
5. 5Volume of Prisms, Pyramids, and ConesCalculate the volume of prisms, pyramids, cones, frustums and spheres
6. 6Volume of Frustums and SpheresCalculate the volume of prisms, pyramids, cones, frustums and spheres
7. 7Volume of Composite SolidsDetermine the volume of composite solids
8. 8Surface Area and Volume of Solids: Real-Life Applications (Part 1)Explore the use of the surface area and volume of solids in real-life situations
9. 9Real-Life Applications of Surface Area and Volume of SolidsExplore the use of the surface area and volume of solids in real-life situations