Curriculum Alignment
- Strand
- 2.0 Measurements and Geometry
- Sub-Strand
- 2.7 Surface Area
- Specific Learning Outcomes
- Explore the use of the surface area and volume of solids in real-life situations
Subsection 2.7.6 Surface Area of Frustums
Teacher Resource 2.7.38.
To assist your teaching, we have prepared lesson resources, aligned with this textbook and the CBC. The Lesson Plan links to syllabus learning outcomes and provides suggest time allocations. The Step-by-Step Guide provides more detailed guidance on how to teach the content, including suggested questions to ask learners, and possible answers.
Learner Experience 2.7.8.
Modelling a frustum using a paper cup or cone-shaped fruit juice glass and calculate its surface area.
\(\textbf{ Materials needed:}\)
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Printable nets of a cone (to cut and create a frustum)Rulers or measuring tapeScissorsTape or glueFormula sheetWorksheets for sketching and calculations
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A frustum is formed when the top part of a cone is cut off parallel to the base.
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Surface area includes:\(\textbf{Curved surface area}\) (side)\(\textbf{Area of both circular bases}\)
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Build the Frustum Model.Take the cone net and cut off the top part (smaller cone) parallel to the base.Assemble the remaining portion to form a frustumAlternatively, use actual paper/plastic cups and measure directly
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Label DimensionsRadius of the larger base (R)Radius of the smaller top base (r)Slant height (l) of the frustum(If not provided, measure the height and use the Pythagorean theorem)
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Calculate Surface Area
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Use the surface area formula:Total Surface Area = \(\pi (R+r)\ell + \pi R^2 + \pi r^2\)
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\(\pi(R+r)\)l: Curved surface\(\pi R^2\text{:}\) Area of bottom base\(\pi r^2\text{:}\) Area of top base
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Example: Given Top radius (r): 3 cm , Bottom radius (R): 5 cm and Slant height (l): 6 cm\begin{align*} \text{Surface Area } =\amp \pi(5\, \text {cm}+3\, \text {cm})(6\, \text {cm}) + \pi(5\, \text {cm})^2 + \pi(3\, \text {cm})^2 \\ = \amp 3.14 \times (8\, \text {cm})\times(6\, \text {cm}) + 3.14 \times 25\, \text {cm} + 3.14 \times 9\, \text {cm}\\ =\amp (3.14 \times \, 48 \text {cm} ) \times(3.14 \times \, 25 \text {cm} ) \times(3.14 \times \, 9 \text {cm} ) \\ = \amp 150.72 \, \text{cm}^2 + 78.50 \, \text{cm}^2 + 28.26\, \text{cm}^2 \\ = \amp257.48\, \text{cm}^2 \end{align*}
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\(\textbf{Study Questions}.\)What would happen to the surface area if the top radius increased?Why is it necessary to measure the slant height, not the vertical height?Can you find any real-life objects shaped like a frustum?
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\(\textbf{Assignment}\)Design their own frustum cups with chosen dimensions.
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Given a full cone, how much surface area is “lost” when the top is cut off?
Key Takeaway 2.7.39.
A frustum is a cone or pyramid is cut parallel to its base, removing the top portion. This results in a truncated shape with two parallel bases one smaller than the other e.g a lampshades, Truncated cones in engineering, buckets, Tunnels, Cooling towers in power plants etc
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Thereby a frustum is the portion of a cone (or pyramid) that remains after the top part is cut off parallel to the base.\(\textbf{Properties of a Frustum}\)
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Two Circular Bases - A frustum has a larger base and a smaller base (both circular).
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Slant Height \(\ell\text{.}\) The distance between the two bases along the side of the frustum.
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Height \(h\) - The vertical distance between the two bases.
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Curved Surface Area (CSA) - The side surface that connects the two bases.
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Total Surface Area (TSA) - The sum of the CSA and the areas of the two circular bases.
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\(\textbf{NOTE:}\) Volume, the space inside the frustum, is calculated using a formula derived from a full cone.
\(\textbf{Types of frustums}\)
\(\textbf{Full cone}\)
Important formulas to note;
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Curved Surface Area (CSA)\begin{equation*} CSA = \pi\, RL \,\text{and} \pi\, rl \, \text{for the smaller cone} \end{equation*}
Example 2.7.40.
Find the surface area of the galvanized iron bucket below.
Solution.
Complete the cone from which the bucket is made, by adding a smaller cone of height \(x \)cm.
From the cocept of similarity and enlargement;
\begin{align*}
\frac{x}{15} = \amp\frac {x + 20 \text{cm} }{20 \text{cm}}\\
20 x = \amp 15x \text{cm} + 300 \text{cm}\\
300 \text{cm} = \amp 20 x - 15 x \\
300 \text{cm} = \amp 5x\\
60 \text{cm} = \amp x
\end{align*}
Surface area of a frustrum = Area of curved surface of bigger cone - Area of curved surface of snaller cone
\(\pi RL - \pi rl\)
\begin{align*}
\text{Surface area (Large)} = \amp \frac {22}{7} \times 20 \text{cm} \times \sqrt{80^2 + 20^2} \\
= \amp 5183.33 \text{cm}\\
\text{Surface area (small)} = \amp \frac {22}{7} \times 15 \text{cm} \times \sqrt{60^2 + 15^2} \\
= \amp 2915.62 \text{cm} \\
\text{Differences in the Surface areas} = \amp 5183.33 \text{cm} - 2915.62 \text{cm}\\
= \amp 2267.71 \text{cm}^2
\end{align*}
Example 2.7.41.
A bucket is in the shape of a frustum of a cone. The top radius is \(14\) cm, the bottom radius is \(6\) cm, and the vertical height of the bucket is \(24\) cm.
Find the total surface area of the bucket (Take \(\pi = 3.142\)).
Solution.
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Use similarity to find the heights of the two cones.From similarity:\begin{align*} \frac{H}{h} = \frac{R}{r}\amp \\ = \amp \frac{14}{6}\\ = \amp \frac{7}{3} \end{align*}So, \(H = \frac{7}{3}h\)Since the frustum height is 24 cm:\(H - h = 24\)\(\frac{7}{3}h - h = 24\)\(\frac{4}{3}h = 24\)\(h = 18 \;\text{ cm}\)Therefore: \(H = 42 \;\text{ cm}\)
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Find slant heights.Large cone:\begin{align*} L = \amp \sqrt{42^2 + 14^2}\\ = \amp \sqrt{1764 + 196}\\ = \amp \sqrt{1960}\\ = \amp 44.29 \;\text{ cm} \end{align*}Small cone:\begin{align*} l = \amp \sqrt{18^2 + 6^2}\\ = \amp \sqrt{324 + 36}\\ = \amp \sqrt{360}\\ = \amp 18.97 \;\text{ cm} \end{align*}
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Curved surface areas.Large cone:\(\pi R L = 3.142 \times 14 \times 44.29 = 1948.7 \;\text{ cm}^2\)Small cone:\(\pi r l = 3.142 \times 6 \times 18.97 = 357.6 \text{ cm}^2\)Curved surface area of frustum:\(1948.7 - 357.6 = 1591.1 \;\text{ cm}^2\)
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Add area of the base.\(\pi r^2 = 3.142 \times 6^2 = 113.1 \;\text{ cm}^2\)Total Surface Area:\(1591.1 + 113.1 = 1704.2 \;\text{ cm}^2\)
Example 2.7.42.
A container is in the shape of a frustum of a rectangular pyramid. The bottom base measures \(20 \text{ cm} \times 12 \text{ cm}\text{,}\) the top base measures \(10 \text{ cm} \times 6 \text{ cm}\text{,}\) and the vertical height is \(15 \text{ cm}\text{.}\)
Find the total surface area of the frustum.
Solution.
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Find the slant heights.Difference in lengths:\(\frac{20 - 10}{2} = 5 \;\text{ cm}\)Slant height (length direction):\begin{align*} \text{Slant height (length direction)} = \amp \sqrt{15^2 + 5^2}\\ = \amp \sqrt{225 + 25}\\ = \amp \sqrt{250}\\ = \amp 15.81 \;\text{ cm} \end{align*}Difference in widths:\(\frac{12 - 6}{2} = 3 \text{ cm}\)\begin{align*} \text{Slant height (width direction)} = \amp \sqrt{15^2 + 3^2}\\ = \amp \sqrt{225 + 9}\\ = \amp \sqrt{234}\\ = \amp 15.33 \;\text{ cm} \end{align*}
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Find areas of trapezoidal faces. Two faces with parallel sides 20 cm and 10 cm:\(\frac{1}{2}(20 + 10)(15.81) = 237.15 \;\text{ cm}^2\)Total for both is \(2 \times 237.15 = 474.3 \;\text{ cm}^2\)Two faces with parallel sides 12 cm and 6 cm:\(\frac{1}{2}(12 + 6)(15.33) = 137.97 \;\text{ cm}^2\)Total for both is \(2 \times 137.97 = 275.94 \;\text{ cm}^2\)
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Add base areas.Bottom base area is \(20 \times 12 = 240 \;\text{ cm}^2\)Top base area is \(10 \times 6 = 60 \;\text{ cm}^2\)
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Total Surface Area.\(474.3 + 275.94 + 240 + 60 = 1050.24 \;\text{ cm}^2\)
\(\textbf{Exercise}\)
1. A frustum of a square pyramid has:
Top square side length: 4 m
Bottom square side length: 6 m
Slant height (along one face): 5 m. Calculate the total surface area of the frustum.
2. A conical frustum has a bottom radius of 6 cm, no top (the top is flat), and a slant height of 10 cm.
i. Explain the difference between the curved surface area and the total surface area of a frustum.
ii. Find only the curved surface area of the frustum.
3. A frustum is formed by cutting a cone with a height of 24 cm into two parts. The smaller cone has a height of 9 cm. If the base radius of the original cone is 16 cm, calculate the total surface area of the frustum.
4. A conical frustum has a bottom radius of 6 cm, no top (the top is flat), and a slant height of 10 cm. Find only the curved surface area of the frustum.
5. If the curved surface area of a frustum is 330 cm², the top radius is 5 cm, and the bottom radius is 10 cm, find the slant height of the frustum.
6. A flower pot is shaped like a frustum of a cone. It has a top radius of 12 cm, a bottom radius of 8 cm, and a slant height of 10 cm.
