Calculating Prism Height And Volume: A Step-by-Step Guide

Hey there, geometry enthusiasts! Today, we're diving into a classic problem involving a right rectangular prism. We'll be using the information provided – the base side length and the diagonal of the prism – to find the height and, ultimately, the volume of this 3D shape. This is a great exercise for understanding how different elements of a prism relate to each other and how to apply the Pythagorean theorem in a 3D context. So, grab your pencils and let's get started!

Understanding the Problem: Right Rectangular Prism

First things first, let's make sure we're all on the same page about what a right rectangular prism is. Imagine a box – that's essentially what we're dealing with. It has six rectangular faces, and all the angles are right angles (90 degrees). The term "right" here is crucial, as it tells us that the sides are perpendicular to the base. The problem gives us a side of the base (5 cm) and the diagonal of the prism (13 cm). Our mission? Find the height and the volume. Remember, the volume of a rectangular prism is how much space it occupies. We'll need to know the height of the prism to calculate the volume. This problem brilliantly connects 2D and 3D geometry, and we will use the Pythagorean theorem, a staple in solving such problems. Understanding this problem allows us to improve our geometrical skills. Let's break this down into manageable chunks.

We are given the side of the base, which is one of the sides of the rectangle that forms the base of the prism. We are also given the diagonal of the prism. This diagonal is a line segment that stretches from one corner of the prism to the opposite corner, passing through the interior. This is like drawing a line from the bottom front left corner to the top back right corner of our box. Knowing these two measurements is enough for us to deduce all other measurements.

To solve this problem, we'll need to use a little bit of spatial reasoning. It is a good idea to visualize the prism and how the given measurements relate to each other. Imagine a rectangular box. The base of the box is a square with sides of 5 cm each. The diagonal of the prism (13 cm) acts as the hypotenuse of a right triangle. One of the legs of this triangle is the height of the prism. The other leg is the diagonal of the base. To find the diagonal of the base, we apply the Pythagorean theorem, and then we can apply the theorem again to find the height. Let's get started. It's all about applying the correct formulas and making sure we understand what each measurement represents within the prism's structure. This problem is a good illustration of how geometry is used in practical applications. Using geometry, we can determine dimensions and spatial relationships. Let's see how to crack this.

Finding the Diagonal of the Base

Alright, let's start by finding the diagonal of the base. Since the base is a square with sides of 5 cm each, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In simpler terms, a² + b² = c², where a and b are the sides of the right triangle, and c is the hypotenuse. In our case, the sides of the square base are a = 5 cm and b = 5 cm, and the diagonal of the base is c. So, let's plug in the numbers: 5² + 5² = c². This simplifies to 25 + 25 = c², which gives us 50 = c². To find c, we take the square root of both sides: √50 ≈ 7.07 cm. So, the diagonal of the base is approximately 7.07 cm.

Remember, the diagonal of the base is a line segment within the base itself, stretching from one corner to the opposite corner of the base. This diagonal, along with the height of the prism and the prism's diagonal (the 13 cm one given in the problem), forms another right triangle. It's like slicing the prism diagonally from one corner on the bottom to the opposite corner on the top. This creates a right triangle where the diagonal of the prism is the hypotenuse, the height is one leg, and the base diagonal is the other leg. With the diagonal of the base calculated, we are one step closer to the final goal.

This step is a crucial part of solving the problem as it connects the 2D properties of the base with the 3D structure of the entire prism. The diagonal of the base serves as a key link in this process, helping us to bridge the gap between two dimensions. Now, we can use this information to find the height of the prism.

So, to recap, we used the Pythagorean theorem to calculate the diagonal of the square base. This diagonal is a necessary intermediate value to determine the height of the prism. Understanding the relationships between the sides and diagonals in a right rectangular prism is fundamental to solving such geometrical problems. Now we are ready to move on to the next step, where we will use the base diagonal and the prism diagonal to find the height of the prism.

Calculating the Height of the Prism

Now, let's find the height of the prism. We've already found the diagonal of the base (approximately 7.07 cm), and we know the diagonal of the prism is 13 cm. We can use the Pythagorean theorem again, this time on the right triangle formed by the height, the diagonal of the base, and the diagonal of the prism. In this case, the diagonal of the prism (13 cm) is the hypotenuse, the diagonal of the base (7.07 cm) is one leg, and the height (let's call it h) is the other leg. So, we have (7.07)² + h² = 13². This simplifies to approximately 50 + h² = 169. Now, subtract 50 from both sides: h² = 119. Taking the square root of both sides, we get h ≈ 10.91 cm. So, the height of the prism is approximately 10.91 cm.

This is the height that extends from the base to the top face, the height is one of the three dimensions that define the prism. To reiterate, we used the diagonal of the prism, the base diagonal, and the Pythagorean theorem to isolate and solve for the height of the prism.

This step perfectly demonstrates the power of the Pythagorean theorem in 3D geometry. It allows us to calculate lengths that are otherwise inaccessible. With the height now in hand, we are ready to determine the volume of the prism, the final piece of the puzzle. We can visualize the height as the vertical distance between the two bases of the prism. Remember that the height, along with the length and the width, are the dimensions needed to calculate the volume. Calculating the height is crucial to finding the volume of the prism. Let's move on to find the volume!

Determining the Volume of the Prism

Finally, let's calculate the volume of the prism. The volume of a right rectangular prism is calculated by multiplying the area of the base by the height. The base is a square with a side length of 5 cm, so the area of the base is 5 cm * 5 cm = 25 cm². We've also calculated the height to be approximately 10.91 cm. Therefore, the volume is 25 cm² * 10.91 cm = 272.75 cm³.

The volume represents the total space enclosed within the prism. It's a measure of how much the prism can contain. So, we have now successfully calculated both the height and the volume. We determined the height using the Pythagorean theorem. We then used the height to compute the volume. Using our derived values, we have our answer to the question.

This means if you were to fill the prism with water, you would need approximately 272.75 cubic centimeters of water. The volume is a measure of the three-dimensional space that the prism occupies. This last step brings everything together. We started with the base side length and the diagonal and ended up calculating the volume of the prism. Understanding the concept of volume is key to solving a variety of geometric problems.

Final Answer and Summary

• Height of the prism: ≈ 10.91 cm
• Volume of the prism: ≈ 272.75 cm³

Congratulations, guys! We've successfully solved the problem. We've found the height of the prism and calculated its volume by applying the Pythagorean theorem twice and using the base area and height. This problem highlights the interconnectedness of different elements in geometry and reinforces the importance of understanding fundamental formulas. Keep practicing, and you'll master these types of problems in no time!