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Grade 10 Mathematics Learners’ Book

INNODEMS

Subsection 2.7.17 Volume of Composite solids

Activity 2.7.22.

Constructing a real-world object using basic solids (cylinder, cube, cone, hemisphere, etc.) and calculate the total volume.
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  • \(\textbf{Materials needed:}\)
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    🔹 Building blocks or 3D shape cut-outs (foam, paper nets, or toy blocks)
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    🔹 Rulers or measuring tapes
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    🔹 Worksheets for drawings and calculations
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    🔹 Volume formula sheet
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  • Review volume formulas for Cube/cuboid, cylinder, cone, sphere, Hemisphere, triangular prisms and pyramids.
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  • Work in groups of atleast four members.
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  • 🔹 Build UP a structure using 2 to 3 shapes. For example
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    🔹 A lighthouse (cylinder + cone)
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    An ice cream cone (cone + hemisphere)
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    🔹 A mailbox (cuboid + half-cylinder)
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    🔹 A robot body (cuboid + cylinder arms + sphere head)
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    🔹 After building up draw the structure, label the parts and measure dimensions. Then alculate the volume of each solid part using the correct formulas.
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    🔹 Then they add all volumes to get the total.
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\(\textbf{Key Takeaway}\)
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A composite solid is a three-dimensional shape made up of two or more simple solids (such as cubes, cylinders, cones, spheres, prisms, and pyramids). To find the volume of a composite solid, the volumes of the individual solids are calculated and then either added or subtracted, depending on the situation.
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  • Identify the simple solids that make up the composite solid
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  • Calculate the volume of each individual solid using the appropriate formula
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  • Add or subtract the volumes i.e:
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    🔹 If the solids are joined together, add their volumes.
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    🔹 If a part of one solid is removed (e.g., a hole), subtract its volume from the total.
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  • Examples of Composite Solids include Cylinder with a Hemisphere on Top, Rectangular Prism with a Cylindrical Hole etc.
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Example 2.7.62.

A rectangular prism has dimensions length = 10 cm, width = 6 cm, and height = 15 cm. A cylindrical hole of radius 2 cm passes vertically through the entire height of the prism. Find the volume of the remaining solid after the hole is removed.
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Solution.
Step 1: Find the Volume of the Rectangular Prism
\begin{align*} V_{\text{prism}} =\amp \text{length} \times \text{width }\times \text{height} \\ = \amp \, 10 \, \text{cm} \times 6 \, \text{cm} \times 15 \, \text{cm}\\ = \amp 900 \, \text{cm}^3 \end{align*}
Step 2: Find the Volume of the Cylindrical Hole
\begin{align*} V_{\text{cylinder}} = \amp \pi r^2h \\ = \amp 3.14 \times \, (2 \,\text{cm})^2 \times 15 \, \text{cm} \\ = \amp 188.4 \, \text{cm}^2 \end{align*}
Step 3: Find the Volume of the Remaining Solid
\begin{align*} V_{\text{Total volume remaining}} = \amp V_{\text{prism}} - V_{\text{cylinder}} \\ = \amp (900 -188.4)\, \text{cm}^3\\ = \amp 711.6 \, \text{cm}^3 \end{align*}
The volume of the remaining solid after the hole is removed is \(711.6 \text{cm}^3\text{.}\)
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\(\textbf{Exercise}\)
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1. A cylinder has a diameter of 14 cm and a height of 15 cm. A hemisphere with the same radius is attached to the top. Find the total volume of the composite solid. (Take \(\pi = \frac{22}{7}\))
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2. A rectangular prism has dimensions 12 cm by 8 cm by 20 cm. A cylindrical hole with a radius of 3 cm is drilled vertically through the entire height of the prism. Find the volume of the remaining prism .
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3. A cylinder has a radius of 5 cm and a height of 12 cm. A cone with the same radius and a height of 9 cm is placed on top. Find the total volume of the solid.
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4. A solid sphere with a radius of 6 cm is completely enclosed in a cube. Find the volume of the space inside the cube but outside the sphere
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5. A rectangular prism has a base of 10 cm by 6 cm and a height of 15 cm. A square pyramid with a base of 10 cm by 10 cm and a height of 8 cm is placed on top of the prism. Find the total volume of the solid.
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Technology 2.7.63.

"Geometry is not just about shapes—it’s about seeing the invisible. With technology, we don’t just see more clearly—we explore more deeply." Adapted for modern mathematics education
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Visit the links below to have interactive exercises that will sharpen your mathematical skills to the uttermost!
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Checkpoint 2.7.64.

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Checkpoint 2.7.65.

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Checkpoint 2.7.66.

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Checkpoint 2.7.67.

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