How to find the volume of a frustum?
Understand the Problem
The question is asking how to calculate the volume of a frustum, which is a geometric shape typically obtained by slicing a cone or a pyramid. To solve this, we will apply the formula for the volume of a frustum, which involves the radii of the two circular ends and the height of the frustum.
Answer
The volume of the frustum is $$ V = \frac{196}{3} \pi \text{ cubic units} $$
Answer for screen readers
The volume of the frustum is:
$$ V = \frac{196}{3} \pi \text{ cubic units} $$
Steps to Solve
- Identify the Formula for the Volume of a Frustum
The volume $V$ of a frustum can be calculated using the formula:
$$ V = \frac{1}{3} \pi h (R^2 + Rr + r^2) $$
where:
- $R$ is the radius of the larger base,
- $r$ is the radius of the smaller base,
- $h$ is the height of the frustum.
- Substitute the Values into the Formula
You'll need the values for $R$, $r$, and $h$. For example, let's assume:
- $R = 5$ units,
- $r = 3$ units,
- $h = 4$ units.
Plugging in these values into the volume formula gives:
$$ V = \frac{1}{3} \pi (4) (5^2 + 5 \cdot 3 + 3^2) $$
- Calculate the Areas and Simplify
First, calculate the terms inside the parentheses:
- $5^2 = 25$
- $5 \cdot 3 = 15$
- $3^2 = 9$
Now add these together:
$$ 25 + 15 + 9 = 49 $$
So the equation becomes:
$$ V = \frac{1}{3} \pi (4) (49) $$
- Calculate the Volume
Now simplify and calculate the volume:
$$ V = \frac{1}{3} \pi \cdot 4 \cdot 49 $$ Calculate $4 \cdot 49 = 196$:
$$ V = \frac{1}{3} \pi \cdot 196 $$
Find the final volume:
$$ V = \frac{196}{3} \pi $$
- Approximate if Necessary
If needed, you can approximate the value using $\pi \approx 3.14$:
$$ V \approx \frac{196}{3} \cdot 3.14 $$
Calculate this to find a decimal approximation.
The volume of the frustum is:
$$ V = \frac{196}{3} \pi \text{ cubic units} $$
More Information
The volume of a frustum is important in various applications, such as calculating the space inside a truncated cone for production in manufacturing or in engineering fields. It's also useful in real-life scenarios, such as designing flower pots or lampshades.
Tips
- Forgetting to square the radii ($R$ and $r$) when applying the formula.
- Mixing up the larger base radius ($R$) with the smaller base radius ($r$).
- Not simplifying the formula properly before performing multiplication, leading to arithmetic mistakes.
