Grade 10 · Core Mathematics · Lesson 7 of 9

Volume of Composite Solids

A Grade 10 Core Mathematics CBE lesson from the sub-strand Surface Area and Volume of Solids. In it, learners decompose composite solids into basic 3D shapes along joining planes.

Strand Measurements and GeometrySub-strand Surface Area and Volume of SolidsLesson code 10CM-27-L740 minutes · 2 slides

KICD learning outcome (d)

Determine the volume of composite solids.

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

Decompose composite solids into basic 3D shapes along joining planes.
Calculate individual component volumes and apply the additive principle to find total volume.

Key vocabulary

Composite solid: A three-dimensional figure constructed by joining two or more basic geometric shapes together.; The kennel structure is a composite solid made of a cuboid and a pyramid.
Additive principle: The rule stating that total volume equals the sum of component shapes' individual volumes.; We applied the additive principle to sum the cuboid and pyramid volumes together.
Compound cross-section: A composite two-dimensional face formed by combining two or more basic geometric plane figures.; The bar's compound cross-section combines a rectangle and a semi-circle of radius 7 cm.

Lesson preview

Slide 1 · Decomposing Composite Solids into Individual ShapesModel · 18 min

Figure 1: A composite solid representing a kennel is split along a dashed joining plane into a rectangular cuboid base and a rectangular pyramid top.

A composite solid is created by joining two or more standard three-dimensional shapes together along a shared plane. To determine the volume of a composite solid, we apply the additive principle: we divide the composite figure into its individual component shapes, calculate the volume of each component using its standard formula, and then sum the individual volumes. Unlike surface area calculations, we do not subtract joined faces when calculating volume because volume measures total internal space.

A composite solid is formed by combining two or more distinct three-dimensional geometric figures along a common boundary.
The total volume of a composite solid equals the sum of the volumes of its component shapes.
Decompose complex structures along joining planes into recognisable 3D solids like prisms, pyramids, cones, or cylinders.
Standard geometric volume formulas are applied independently to each component shape before summing their calculated values.

Worked example. Calculate the volume of the cuboid base:

V_{1} = length \times width \times height = 6 \times 4 \times 3 = 72 m^{3}

. Calculate the volume of the pyramid top:

V_{2} = \frac{1}{3} \times base area \times height = \frac{1}{3} \times (6 \times 4) \times 2.5 = 20 m^{3}

. Add the component volumes:

V_{total} = V_{1} + V_{2} = 72 + 20 = 92 m^{3}

. Answer: The total volume of the grain container is $92 m^{3}$ ..

Check question: A water reservoir model consists of a cylinder of radius $7 cm$ and height $5 cm$ , capped with a cone of the same radius $7 cm$ and height $6 cm$ . Taking $π = \frac{22}{7}$ , calculate the total volume of the structure.

$154 {cm}^{3}$
$770 {cm}^{3}$
$1078 {cm}^{3}$
$1694 {cm}^{3}$

Show answer

$1078 {cm}^{3}$ . Volume of the cylinder $V_{1} = π r^{2} h_{1} = \frac{22}{7} \times 7^{2} \times 5 = 770 {cm}^{3}$ . Volume of the cone $V_{2} = \frac{1}{3} π r^{2} h_{2} = \frac{1}{3} \times \frac{22}{7} \times 7^{2} \times 6 = 308 {cm}^{3}$ . Total volume $V = 770 + 308 = 1078 {cm}^{3}$ .

Slide 2 · Calculating Volumes of Complex Composite StructuresPractice · 22 min

Figure 2: A composite prism representing a metallic bar features a shaded cross-section made of a rectangle and a semi-circle, with its length indicated.

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Teacher notes, SNE adaptations and assessment checks
Homework and the other 8 lessons in this sub-strand

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Volume of Composite Solids

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

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 9 lessons in Surface Area and Volume of Solids