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Subsection 2.8.5 Volume of Cones

Learner Experience 2.8.34.

Constructing a cone.
  1. Materials Needed
    Sheets of paper or cardboard
    Scissors, glue/tape, and rulers
    A cylinder (e.g., cup or bottle) for comparison
    • "How can we turn this into a cone?"
      Take a piece of paper and \(\textbf{cut a circle}\) any radius.
    • \(\textbf{Cut out a sector and roll}\) the remaining part into a cone shape.
    • Measure the radius and height of their cones.
    • Calculate the volume using the formula .
    • Equally, we can try this activity using; Empty Ice cream cones, A cylindrical cup of the \(\textbf{same height}\) and \(\textbf{base}\) as the cone and water.
      Steps:
    • Fill the cone with water and pour it into the cylinder severally until the cylinder is full.
    • How many cones of rice will fill the cylinder?
    • \(\textbf{Notice that it takes exactly 3 full cones to fill the cylinder}\text{.}\)
      This is why the formula includes \(\frac{\textbf{1}}{\textbf{3}} \text{!}\)
      Mathematical Insight
      ✔ This shows the formula:
      \begin{equation*} V_\text{cone} = \frac{1}{3} V_\text{cylinder} = \frac{1}{3} \pi r^2h \end{equation*}
      ✔ The cone is one-third of the volume of a cylinder with the same base and height.

Key Takeaway 2.8.35.

What is a Cone? A cone is a three-dimensional solid with a circular base and a curved surface that tapers to a point called the apex or vertex.
Formula:
Volume of a Cone \(= \frac{1}{3} \times r^2 \times h\)
Where:
  1. \(r\) is the radius of the circular base
  2. \(h\) is the height of the cone (perpendicular distance from base to apex)
  3. \(\displaystyle \pi \approx 3.14 \text{or} \frac{22}{7} \)
Why \(\frac{1}{3}\)
Three cones with the same base and height fit exactly inside a cylinder with the same base and height. Therefore, the volume of a cone is one-third the volume of a cylinder.
Relationship to Cylinder:
  1. \(V\) cylinder \(= \frac{1}{3} \times V\) cylinder
  2. \(V\) cylinder \(= \pi \times r^2 \times h\)
  3. Therefore, \(V\) cone \(= \frac{1}{3} \times \pi \times r^2 \times h\)
Scaffolding Strategies to Address Misconceptions:
Misconception Clarification
A cone is the same as a cylinder No, a cone has a circular base and tapers to a point, while a cylinder has two parallel circular bases
The volume formula is \(\pi \times r^2 \times h\) No, you must multiply by \(\frac{1}{3}\)
The height is the slant height No, the height is the perpendicular distance from the base to the apex, not the slant height along the curved surface
I can use the diameter instead of the radius No, the formula uses the radius. If you have the diameter, divide by 2 to get the radius

Example 2.8.36.

Find the volume of the following cone (correct to 1 decimal place):
Solution.
Step 1: Find the area of the base .
\begin{align*} \text{Area of a Circle} = \amp \pi r^2 \\ = \amp \frac{22}{7} \times 14\, \text{cm} \times 14 \, \text{cm} \\ = \amp 616{cm}^2 \end{align*}
Step 2: Calculate the volume
\begin{align*} \text{V} =\amp \frac{1}{3} \times \pi r^2\times H \\ = \amp 616\, \text{cm}^2 \times 28\, \text{cm} \\ = 17, 248\, \text{cm}^3 \amp \end{align*}

Example 2.8.37.

The slant height of a cone is \(13\) cm and the radius of its base is \(5\) cm.
Find:
  1. The vertical height of the cone.
  2. The volume of the cone (Give your answer in terms of \(\pi\) and then correct to two decimal places).
Solution.
  1. In a cone, the radius, height and slant height form a right-angled triangle.
    \(l^2 = r^2 + h^2\)
    Substitute \(l = 13\) and \(r = 5\)
    \(13^2 = 5^2 + h^2\)
    \(169 = 25 + h^2\)
    \(h^2 = 144\)
    \(h = 12 \;\text{ cm}\)
  2. Use the volume formula
    \begin{align*} V =\amp \frac{1}{3}\pi r^2 h\\ = \amp \frac{1}{3}\pi (5)^2 (12)\\ = \amp \frac{1}{3}\pi (25)(12)\\ = \amp \frac{1}{3}\pi (300)\\ V = \amp 100\pi \end{align*}
    Therefore, \(V = 100\pi \;\text{ cm}^3\)
    Using \(\pi \approx 3.142\)
    \(V \approx 314.16 \;\text{ cm}^3\)

Checkpoint 2.8.38.

Checkpoint 2.8.39.

Exercises Exercises

1.

A cone has a radius of 12 cm and a height of 18 cm. Calculate the volume of the cone.
Answer.
\(2714.34 \;\text{cm}^3\)

2.

An ice cream cone has a radius of 3 cm and a height of 8 cm. Estimate how much ice cream it can hold?
Answer.
\(75.4 \;\text{cm}^3\)

3.

Two cones have the same height of 84 cm but different radii. The first cone has a radius of 14 cm, and the second cone has a radius of 42 cm. Calculate and compare the volumes of the two cones. Which one has a larger volume?
Answer.
\(V_1 = 17241.06 \;\text{cm}^3\)
\(V_2 = 155169.55 \;\text{cm}^3\)
The second cone has the larger volume.

4.

A cone-shaped funnel has a radius of 9 cm and a height of 18 cm.How much water can the funnel hold? (Leave your answer in cubic meters.)
Answer.
\(1.53 \times 10^{-3}\;\text{m}^3\)

5.

A cone has an outer radius of 7 cm and an inner radius of 5 cm. The height of the cone is 12 cm.Calculate the volume of the hollow cone.
Answer.
\(301.59 \;\text{cm}^3\)
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