Surface Area of Frustums, Spheres, and Hemispheres

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

1. Calculate the surface area of pyramid frustums and conical frustums using face summation and similar triangles.
2. Derive and apply the formula $A=4\pi r^2$ for the surface area of a sphere.
3. Calculate the surface area of solid hemispheres and open hemispherical bowls.

Key vocabulary

Frustum

The remaining solid portion formed when the top of a pyramid or cone is cut off.

The metal grain hopper forms a frustum of a pyramid with two square faces.

Slant height

The diagonal distance along the outer curved edge from base perimeter to apex or top edge.

The frustum of a cone has a slant height of $20\text{ cm}$ along its outer side.

Sphere

A perfectly round three-dimensional solid figure where all surface points are equidistant from centre.

A football or shot put spherical solid has a surface area calculated using $4\pi r^2$.

Hemisphere

Half of a sphere bounded by a circular plane and a curved hemispherical surface.

A solid hemisphere of radius $0.35\text{ m}$ has surface area equal to $3\pi r^2$.

Lesson preview

Slide 1 · Surface Area of Pyramid FrustumsOpening · 10 min

A frustum of a pyramid is created when the top portion of a pyramid is sliced off by a plane parallel to its base. Its total surface area comprises the areas of two parallel polygonal faces (the top and bottom bases) and the lateral trapezoidal faces connecting them. For a square pyramid frustum, adding the top square, base square, and four congruent trapezia gives the total area.

• A frustum of a pyramid consists of top and bottom parallel polygonal faces plus slant trapezoidal faces.
• The total surface area equals the sum of the two base areas and all surrounding trapezoidal side faces.
• The area of each trapezoidal side face is calculated using $\frac{1}{2}(a+b)h$ with slant face height $h$.
• Summing all constituent face areas systematically yields the exact surface area of the pyramid frustum.

Worked example. Calculate top square face area: $A_1=20\times 20=400{\text{ cm}}^2$. Calculate bottom square face area: $A_2=40\times 40=1600{\text{ cm}}^2$. Calculate area of one trapezoidal side face: $A_{\text{trap}}=\frac{1}{2}\times (20+40)\times 25=30\times 25=750{\text{ cm}}^2$. Calculate area of four trapezoidal faces: $4\times 750=3000{\text{ cm}}^2$. Sum all face areas: $\text{Total Surface Area}=400+1600+3000=5000{\text{ cm}}^2$. Answer: $5000{\text{ cm}}^2$.

Check question: A pyramid frustum has square ends of side lengths $10\text{ cm}$ and $6\text{ cm}$, and a slant face height of $12\text{ cm}$. What is its total surface area?

1. $384{\text{ cm}}^2$
2. $484{\text{ cm}}^2$
3. $520{\text{ cm}}^2$
4. $904{\text{ cm}}^2$

Show answer

$520{\text{ cm}}^2$. Top square area is $6^2=36{\text{ cm}}^2$. Bottom square area is ${10}^2=100{\text{ cm}}^2$. The area of the four trapezoidal side faces is $4\times \frac{1}{2}(6+10)\times 12=384{\text{ cm}}^2$. Total surface area is $36+100+384=520{\text{ cm}}^2$.

Slide 2 · Surface Area of Conical FrustumsModel · 10 min

Teach the full lesson on the projector

This lesson, and every other Grade 1–10 CBE lesson, comes ready to project with the EAP ClassKit. It runs without electricity or internet.

• 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