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Grade 10 Mathematics Learners’ Book

INNODEMS

Subsection 2.7.16 Volume of a Frustum

Activity 2.7.21.

Frustum Volume Lab
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Use a bucket-shaped container to model a frustum and measure how much water it holds—connecting real-world experience with the math behind volume.
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  1. \(\textbf{ Materials Needed:}\)
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    🔹 Calculator
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    🔹 Rulers or measuring tapes
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    🔹 Water and a measuring jug (optional but powerful visual!)
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    🔹 Real plastic buckets, measuring cups, or flowerpots (frustum-shaped)
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    🔹 Worksheets for dimensions and calculations
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  2. 🔹 Review the volume formula of a cone and note that the frustum is a cone with the top sliced off.
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    🔹 Observe real-life frustums: buckets, lampshades, party hats cut short, flower pots, juice glasses, etc.
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  3. Students work in pairs or small groups. They measure:
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    Diameter (then radius) of top opening: \(\textbf{R}\)
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    Diameter (then radius) of bottom: \(\textbf{r}\)
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    Height of the container: \(\textbf{h}\)
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    Measure and record the dimensions label the top radius R and bottom radius r and the height h in your worksheet and record all measurements in cm..
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  4. Fill the bucket with water and pour it into a measuring jug to find it’s actual volume in liters. Then calvulate using the formula and find the volume’s capacity in liters.
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\(\textbf{Key Takeaway}\)
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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 lampshade
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\(\textbf{Types of frustums}\)
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Example 2.7.61.

A frustum of a cone has a top radius of 4 cm, a bottom radius of 8 cm, and a height of 10 cm.
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a. Find the slant height of the frustum.
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b. Find the volume of the frustum.
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Solution.
Using the Pythagoras theorem, the slant height \(\ell\) is given by:
\begin{align*} \ell = \amp \sqrt{8^2 + 10^2}\\ = \amp 12.80 \, \text{cm} \end{align*}
Slant height \(\ell = 12.80\, \text{cm}\)
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Finding the volume. \(V = \frac{1}{3}\pi h (r^2+ R^2 + Rr)\)
\begin{align*} V = \amp \frac{1}{3} \times \frac{22}{7} \times 10\, \text{cm} \times (4^2 + 8^2 + (8\times 4))\, \text{cm} \\ = \amp \frac {22}{21} \times 10\, \text{cm} \times (16\, \text{cm}+64\, \text{cm}+32\, \text{cm}) \\ = \amp \frac{220}{21} \times 112\, \text{cm}^2\\ = \amp 1173.33 \, \text{cm}^3 \end{align*}
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\(\textbf{Exercise}\)
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1. Shell petrol station has the shape of an inverted right circular cone of the radius 7 m and height 15m dug underground in the shape of the figure shown below. Calculate the volume of petrol it holds when full in litre.
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2. A frustum of a cone has a base radius of 8 cm, a top radius of 5 cm, and a height of 12 cm. Find its volume.
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3. A square pyramid is cut into a frustum, where the original height was 18 cm, and the truncated top part has a height of 6 cm. The base side length is 12 cm, and the top side length is 6 cm. Find the volume of the frustum.
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4. A frustum of a cone has a volume of 900 cm³, a base radius of 10 cm, and a top radius of 6 cm. Find its height.
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5. An A improvised jerrycan shaped like a frustum of a cone with a top radius of 20 cm, a bottom radius of 15 cm, and a height of 30 cm. How many liters of water can the bucket hold?
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