Problem 12 - Thomas Calculus in SI Units

Question

Find the volume of the given pyramid, which has a square base of area 9 and height 5.

Step-by-Step Solution

Answer

The volume of the pyramid is 15 cubic units.

Step 1: Understand the problem

We need to find the volume of a square pyramid. The problem provides us with the area of the base and the height of the pyramid. We will use the formula for the volume of a pyramid.

Step 2: Identify the formula for volume

The volume \( V \) of a pyramid is given by the formula \( V = \frac{1}{3} imes ext{Base Area} imes ext{Height} \). This formula applies to pyramids regardless of the shape of the base.

Step 3: Substitute given values into the formula

The base area is given as 9 and the height is 5. Substitute these values into the formula: \( V = \frac{1}{3} imes 9 imes 5 \).

Step 4: Calculate the volume

First, multiply the base area by the height: \( 9 \times 5 = 45 \). Then multiply this result by \( \frac{1}{3} \): \( \frac{1}{3} imes 45 = 15 \).

Step 5: Conclusion

The volume of the pyramid is 15 cubic units.

Key Concepts

Square Pyramid

The square pyramid is one of the simplest forms of pyramids. It has a square base and four triangular sides or faces converging to a point called the apex. To visualize it better, imagine a tent supported by four stakes placed at the edges of a square.
In architecture and nature, square pyramids are found in structures like the Great Pyramid of Giza or in crystals that have a pyramidal shape.

• The base of the pyramid is a geometric square, meaning all four sides of the base are equal in length.
• The apex is directly above the center of the square base, making the pyramid symmetrical.
• The sides, commonly referred to as faces, are triangles; making it a solid with 5 faces in total.

Understanding the structure of a square pyramid is essential as it forms the foundation for calculating various attributes such as surface area and volume.

Volume Formula

When it comes to calculating the volume of a pyramid, there's a simple formula that will help you. The volume of any pyramid is given by this general formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
The formula reflects the fact that a pyramid occupies only a third of the volume of a prism with the same base and height.

• The base area refers to the area of the base shape, which in our case, is a square.
• The height is the perpendicular distance from the base to the apex.

For our square pyramid, we have a base area of 9 square units and a height of 5 units. Plugging these values into the formula, we calculate the volume as:
\[ V = \frac{1}{3} \times 9 \times 5 = 15 \text{ cubic units} \]This formula is powerful because it applies to any shape of the base. Whether the base is triangular, square, or hexagonal, this same formula can be adapted.

Geometric Solids

Geometric solids, also known as three-dimensional shapes, are figures with width, height, and depth. They have surface area and volume — two measurements essential for understanding their properties and differences.
Geometric solids are everywhere, from the spheres you find in sports balls to the cuboids like cereal boxes.

• A pyramid, such as the square pyramid, is one type of geometric solid which is categorized as a polyhedron. It has flat polygonal faces and straight edges.
• Other examples of geometric solids include cubes, cylinders, spheres, and cones.
• Understanding geometric solids helps in various applications: from calculating the amount of space a container can hold, to engineering and architectural planning.

The volume of these solids, including pyramids, can inform anything from how much a structure will weigh to how much liquid it can contain, making these calculations particularly significant in practical scenarios.