Cube Cross-Section Area: Volume 27 Cm³, Angles 30° & 60°
Hey guys! Ever wondered how to calculate the area of a cross-section inside a cube? It might sound intimidating, but trust me, we'll break it down into easy-to-understand steps. In this article, we're tackling a classic geometry problem: finding the area of a cross-section of a cube when you know its volume and the angle of the cutting plane. We'll specifically look at a cube with a volume of 27 cm³ and cross-sections inclined at 30° and 60°. So, grab your thinking caps, and let's dive in!
Understanding the Basics
Before we jump into the calculations, let's make sure we're all on the same page with the fundamentals. The key here is understanding the relationships between the cube's volume, its side length, the diagonals of its faces, and the angles involved in the cross-section. Don't worry if these concepts seem a bit jumbled now; we'll clarify everything as we go.
- Volume of a Cube: The volume (V) of a cube is calculated by cubing the length of its side (a): V = a³. This means if we know the volume, we can easily find the side length by taking the cube root. In our case, the volume is 27 cm³, so we'll be using this to determine the side length shortly.
- Cube Diagonals: A cube has two main types of diagonals we need to consider: face diagonals and space diagonals. A face diagonal runs across a square face of the cube, while a space diagonal runs from one corner of the cube to the opposite corner through the interior. For this problem, we are primarily concerned with the face diagonal, which we'll use as a reference line for our cross-section plane. The length of a face diagonal can be found using the Pythagorean theorem, as it forms the hypotenuse of a right triangle with two sides of the cube as legs. If the side length is 'a', the face diagonal is a√2.
- Cross-Section Plane: The problem describes a cross-section created by a plane. Think of this as slicing the cube with a flat surface. The shape of this slice, and thus its area, depends on the angle at which the plane intersects the cube. We're given two specific angles: 30° and 60°. These angles refer to the inclination of the plane relative to the base of the cube, which significantly affects the shape and size of the resulting cross-section.
- Trigonometry: Understanding trigonometric functions like sine, cosine, and tangent is crucial when dealing with angles. These functions relate angles to the ratios of sides in right triangles. In our problem, we'll use trigonometric relationships to find the dimensions of the cross-section, which will ultimately allow us to calculate its area. Specifically, we'll leverage the given angles (30° and 60°) to determine the heights and lengths within the cross-sectional shape. Make sure you remember your special right triangles (30-60-90) and their side ratios!
Problem Setup: Visualizing the Cross-Section
Okay, with the basics covered, let's visualize the problem. This is super important, guys, because geometry is all about seeing the shapes and their relationships. Imagine our cube sitting on a table. We have a plane slicing through it, but not just any slice! This plane has to meet two key conditions:
- It contains one of the diagonals of the cube's base. Think of this diagonal as an anchor line for our plane.
- It's inclined at either 30° or 60° to the base. This is the angle at which the plane "tilts" relative to the table.
When this plane slices through the cube, it creates a cross-sectional shape. This shape will be a rectangle because the cutting plane intersects two opposite faces of the cube parallel to each other. To find the area of this rectangle, we need its length and width. The length will be the diagonal of the base (which we already discussed), and the width will depend on the angle of inclination. This is where our trigonometry skills will come in handy.
So, visualizing this setup is the first big step. Imagine the diagonal slicing across the base, and then picture the plane tilting upwards at the specified angle. This mental image will guide us as we move into the calculations.
Step-by-Step Calculation: Volume to Side Length
Let's start with what we know: the volume of the cube is 27 cm³. This is our launching pad for finding the side length. As we discussed earlier, the volume (V) of a cube is related to its side length (a) by the formula:
V = a³
To find the side length, we simply take the cube root of the volume:
a = ∛V
In our case:
a = ∛27 cm³
What number, when multiplied by itself three times, equals 27? That's right, it's 3!
a = 3 cm
So, we've successfully found that the side length of our cube is 3 cm. This is a crucial piece of information because it's the foundation for all our subsequent calculations. With the side length in hand, we can now determine the length of the base diagonal and eventually the dimensions of the cross-sectional rectangle.
Finding the Base Diagonal
Now that we know the side length of the cube (3 cm), let's calculate the length of the diagonal of the base. Remember, the base of the cube is a square, and the diagonal of a square divides it into two right-angled triangles. We can use the Pythagorean theorem to find the length of the diagonal.
Let's call the length of the base diagonal 'd'. According to the Pythagorean theorem:
d² = a² + a²
Where 'a' is the side length of the square.
Substituting the value of 'a' (3 cm):
d² = 3² + 3² d² = 9 + 9 d² = 18
Now, we take the square root of both sides to find 'd':
d = √18 cm
We can simplify √18 by factoring out the largest perfect square, which is 9:
d = √(9 * 2) cm d = 3√2 cm
Excellent! We've found the length of the base diagonal, which is 3√2 cm. This diagonal will serve as one side of our cross-sectional rectangle, so we're one step closer to finding its area. Remember this value, guys; we'll be using it in the next steps.
Calculating the Cross-Section Width: The 30° Case
This is where the fun with trigonometry begins! We're now going to calculate the width of the cross-sectional rectangle when the cutting plane is inclined at 30° to the base. Imagine a right-angled triangle formed by:
- The width of the cross-section (which is what we want to find).
- A portion of the cube's edge (which we know is 3 cm).
- A line running along the cutting plane.
The angle between the base (the cube's edge) and the line along the cutting plane is 30°. The width of the cross-section is opposite to this angle, and the cube's edge is adjacent to it. This means we'll be using the tangent function, which relates the opposite and adjacent sides of a right-angled triangle:
tan(angle) = opposite / adjacent
In our case:
tan(30°) = width / 3 cm
We know that tan(30°) = 1/√3, so:
1/√3 = width / 3 cm
To find the width, we multiply both sides by 3 cm:
width = (3 cm) / √3
To rationalize the denominator, we multiply both the numerator and denominator by √3:
width = (3√3 cm) / 3
width = √3 cm
Great! The width of the cross-section when the angle is 30° is √3 cm. Now we have both the length (3√2 cm) and the width (√3 cm), so we can calculate the area.
Calculating the Cross-Section Area: 30° Inclination
With the length (3√2 cm) and width (√3 cm) in hand, finding the area of the rectangular cross-section is straightforward. The area of a rectangle is simply:
Area = length * width
Substituting our values:
Area = (3√2 cm) * (√3 cm) Area = 3√(2 * 3) cm² Area = 3√6 cm²
So, the area of the cross-section when the plane is inclined at 30° to the base is 3√6 cm². That's one part of our problem solved!
Calculating the Cross-Section Width: The 60° Case
Now, let's tackle the second part of the problem: calculating the cross-section width when the plane is inclined at 60° to the base. We'll use a similar approach to the 30° case, relying on trigonometry and the right-angled triangle formed within the cube.
Again, we have a right-angled triangle with:
- The width of the cross-section (our unknown).
- A portion of the cube's edge (3 cm).
- A line along the cutting plane.
This time, the angle between the base and the cutting plane is 60°. We're still dealing with the opposite (width) and adjacent (cube's edge) sides, so we'll use the tangent function again:
tan(angle) = opposite / adjacent
In this case:
tan(60°) = width / 3 cm
We know that tan(60°) = √3, so:
√3 = width / 3 cm
To find the width, we multiply both sides by 3 cm:
width = 3√3 cm
Excellent! The width of the cross-section when the angle is 60° is 3√3 cm. Notice how this is different from the 30° case – the steeper angle results in a wider cross-section. We're now ready to calculate the area for this inclination.
Calculating the Cross-Section Area: 60° Inclination
We've got the length (3√2 cm) and the width (3√3 cm) for the 60° inclination. Let's plug these values into the area formula:
Area = length * width Area = (3√2 cm) * (3√3 cm) Area = 9√(2 * 3) cm² Area = 9√6 cm²
And there you have it! The area of the cross-section when the plane is inclined at 60° to the base is 9√6 cm². This completes our calculations for both angles.
Final Results and Key Takeaways
Alright, guys, we've successfully navigated this geometry challenge! Let's summarize our findings:
- For a cutting plane inclined at 30° to the base, the area of the cross-section is 3√6 cm².
- For a cutting plane inclined at 60° to the base, the area of the cross-section is 9√6 cm².
Notice that the area of the cross-section is significantly larger for the 60° inclination compared to the 30° inclination. This makes sense intuitively – a steeper angle will result in a larger slice through the cube.
This problem highlights the power of combining geometry and trigonometry. By visualizing the shapes, understanding the relationships between sides and angles, and applying the appropriate trigonometric functions, we can solve complex problems. Remember, guys, practice makes perfect, so try working through similar problems to solidify your understanding.