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Grade 10 Mathematics Learners’ Book

INNODEMS

Subsection 2.7.9 Volume of Solids.

Activity 2.7.12.

\(\textbf{Materials Needed:}\)
  1. Transparent plastic containers shaped as: Cube or rectangular prism, Cylinder, Cone, Pyramid (square or triangular base) and Spheres (or hemispheres)
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  2. Sand, water, or rice (as filling material)
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  3. Measuring cup (with volume in millilitres or cm³)
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  4. Large tray or basin (to catch spills)
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  5. Worksheet for recording observations and making predictions
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  • Take different empty containers and weigh them. Which shape do you think holds the most? or the least? Rank them from largest to smallest volume. Let them share and discuss predictions in small groups
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  • Fill each of them separately and Compare
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    Start with the cylinder and cone with the same height. Fill the cone with sand/water and pour it into the cylinder
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  • How many cones fill the cylinder?
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    \begin{align*} 3\, \amp\text{That is the cone is} \frac{1}{3} \text{a cylinder}. \end{align*}
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  • What is the volume of a cylinder? \(V = \pi r^2\)
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    If a cone is \(\frac{1}{3} \, \text{a cylinder then it's volume will be; } \frac{1}{3} \pi r^2\)
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  • Do the same with a pyramid and a matching prism (same base and height). How many pyramids fill the prism?
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    3 pyramids fill tthe prism.
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  • What is the volume of a pyramid?
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    \(v = \frac{1}{2} \times \text{base area} \times h\)
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  • Try filling the sphere into the cylinder (if you have a hemisphere: about 2 hemispheres = 1 sphere)
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  • Compare volumes visually and discuss the differences with your classmates.
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  • For cube or prism, measure directly with a ruler and calculate using \(V = \text{length} \times \text{width} \times \text{height}\)
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  • Measure the dimensions and use their measuring cups to check how much each container holds. They then calculate the actual volume using the formulas and compare their estimates and results.
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\(\textbf{Study Questions}\)
  1. What patterns do you notice between shapes that have the same base and height?
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  2. Why do you think cones and pyramids have a \(\frac{1}{3}\) in their volume formula?
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  3. Which shapes are most efficient at holding volume?
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\(\textbf{Extended Activity}\)
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Design a container that holds exactly 500 cm³ using any shape. Estimate the volume of irregular solids using displacement.
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\(\textbf{Volume of a Cube}\)
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\(\textbf{Volume}\) is the geometric space occupied by an object, or the contents of an object. It is measured in cubic units.
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Computation of a volume is achieved by multiplying the area of the base of the solid by the height of the solid
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Let us proove this from the following examples.
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Example 2.7.31.

Find the volume of the following cube whose side is \(5\, \text{cm}.\)
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Solution.
Step 1: Find the area of the base
\begin{align*} \text{Area of square} = \amp S^2\\ = \amp 5^2\\ = \amp 25 \, \text{cm}^2 \end{align*}
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Step 2: Multiply the area of the base by the height of the solid to find the volume
\begin{align*} \text{volume} = \amp \text {areaof base} \times \text{height}\\ = \amp 25 \, \text{cm}^2 \times 5 \, \text{cm}\\ = \amp 125 \, \text{cm}^3 \end{align*}
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The volume of the cube is 125 \(125 \, \text{cm}^3\text{.}\)
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Example 2.7.32.

Finding the volume of a triangular prism.
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Solution.
Step 1: Find the area of the base
\begin{align*} \textbf{area of triangle =}\amp \frac{1}{2}b \times h\\ = \amp \frac{1}{2} \times 9\, \text{cm} \times 12 \, \text{cm} \\ = \amp 54 \, \text{cm}^2 \end{align*}
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Step 2: Multiply the area of the base by the height of the solid to find the volume
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\begin{align*} \text{Volume} = \amp \text{Base Area} \times \text{height}\\ = \amp \frac{1}{2}b \times h \times H\\ = \amp 54 \, \text{cm}^2\times 25\, \text{cm} \\ = \amp 1350 \, \text{cm}^3 \end{align*}
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The volume of the triangular prism is \(1350 \, \text{cm}^3\)
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Example 2.7.33.

Find the volume of the following cylinder using \(\pi = 3.142\text{.}\) Leave your answer (correct to 2 decimal place):
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Solution.
Step 1: Find the area of the base
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\begin{align*} \text {area of circle } = \amp \pi r^2 \\ = \amp \pi(8)^2\\ = \amp \pi 64\,\text{cm}^2\\ = \amp 3,142 \times 64\,\text{cm}^2 \\ = \amp 201.088 \,\text{cm}^2 \end{align*}
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Step 2: Multiply the area of the base by the height of the solid to find the volume
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\begin{align*} \text{Volume} = \amp \text{Base area} \times \text{height}\\ = \amp \pi r^2 \times h \\ = \amp201.088 \, \text{cm}^2 \times 20\, \text{cm} \\ = \amp 4021.76 \, \text{cm}^3 \end{align*}
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The volume of the cylinder is \(4021.76 \, \text{cm}^3 \)
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\(\textbf{Exercise}\)
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1. A cube has a side length of 5 cm. Find its volume.
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2. The cube below has a volume of 343 cm³. Find the length of one side of the cube.
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3. If the side length of a cube is doubled, by what factor does the volume increase?
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4. A big cube is made by stacking 8 smaller identical cubes together. If the volume of each small cube is 27 cm³, find the volume of the big cube.
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5. (a) A cubical water tank has a side length of 3 m. How many liters of water can it hold when full?
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(b) A cubical water tank is 2 m on each side. If it is filled completely, how many liters of water can it hold?
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6. A sugar cube has a side length of 1 cm. If 1,000 sugar cubes are stacked together to form a larger cube, find the volume of the larger cube.
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7. A freezer contains 200 ice cubes, each shaped like a cube with a side length of 4 cm. Find the total volume of ice inside the freezer.
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8. A cubical shipping container has a side length of 2.5 m. Find the total volume of cargo space inside the container.
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