Problem 31
Question
a. Find the volume enclosed by the pyramidal roof on a square tower. Take the base as \(22.0 \mathrm{ft}\) on a side and the height as \(24.5 \mathrm{ft},\) and ignore the overhang. b. Find the lateral area of the roof.
Step-by-Step Solution
Verified Answer
Volume of the Pyramid: \( V = \frac{1}{3} \times 484.0 \times 24.5 \) cubic feet. Lateral Area of the Pyramid: \( LA = 4 \times \frac{1}{2} \times 22.0 \times l \) square feet, where \( l \) is the slant height.
1Step 1: Determine the Volume of the Pyramid
To find the volume of the pyramidal roof, use the formula for the volume of a pyramid, which is \( V = \frac{1}{3}Bh \), where \( B \) is the area of the base and \( h \) is the height. The base is a square with a side length of \( 22.0 \) ft, so \( B = 22.0 \times 22.0 = 484.0 \) square feet. Then plug in the given height \( h = 24.5 \) ft to calculate the volume.
2Step 2: Calculate the Volume of the Pyramid
Substitute the known values into the volume formula to get \( V = \frac{1}{3} \times 484.0 \times 24.5 \). Perform the multiplication and division to find the volume.
3Step 3: Determine the Lateral Area of the Pyramid
For a square pyramid, the lateral area \( LA \) can be found using \( LA = \sum (\text{Area of each triangular face}) \). Each triangular face has a base equal to the side of the square base, \( 22.0 \) ft, and a slant height, \( l \), which must be calculated using the Pythagorean theorem in the form: \( l = \sqrt{\frac{1}{2} \times 22.0^{2} + 24.5^{2}} \).
4Step 4: Calculate the Lateral Area of the Pyramid
First, calculate the slant height \( l \) to then find the area of one of the triangular faces. Multiply the area by 4, as there are four identical triangular faces, to get the total lateral area.
Key Concepts
Volume of a PyramidLateral Area of a PyramidPythagorean TheoremSquare Pyramid
Volume of a Pyramid
Finding the volume of a pyramid is a fundamental geometric problem that allows us to determine the capacity of pyramidal structures. In the case of a square pyramid, the formula to calculate the volume is relatively straightforward. Let's start by using the given formula for the volume of a pyramid:
\( V = \frac{1}{3}Bh \).
Here, \( B \) is the area of the base, which is a square, and \( h \) is the height of the pyramid. For the pyramid in the exercise, the length of one side of the square base is \( 22.0 \) ft. To find \( B \), we square this side length:\[ B = 22.0 \times 22.0 = 484.0 \text{ square feet} \].
After calculating the base area, the volume can be found by substituting \( B \) and the height \( h = 24.5 \) ft into the formula and completing the necessary calculations. The simplicity of this formula makes it an essential tool for anyone studying geometry or involved in projects where space optimisation is critical.
\( V = \frac{1}{3}Bh \).
Here, \( B \) is the area of the base, which is a square, and \( h \) is the height of the pyramid. For the pyramid in the exercise, the length of one side of the square base is \( 22.0 \) ft. To find \( B \), we square this side length:\[ B = 22.0 \times 22.0 = 484.0 \text{ square feet} \].
After calculating the base area, the volume can be found by substituting \( B \) and the height \( h = 24.5 \) ft into the formula and completing the necessary calculations. The simplicity of this formula makes it an essential tool for anyone studying geometry or involved in projects where space optimisation is critical.
Lateral Area of a Pyramid
Understanding the concept of lateral area is important when dealing with pyramids—especially if we're interested in materials needed to cover the sides or the surface area. The lateral area of a square pyramid specifically refers to the total area of the four triangular faces, not including the base. To calculate the lateral area \( LA \), we sum up the areas of these triangular faces.
Using the exercise as an example, with each triangular face having a base of \( 22.0 \) ft (the side of the square base), we would need to know the slant height \( l \) to calculate the area of a single triangle. We can find \( l \) using the Pythagorean theorem, which leads us to the next important concept in geometry: the Pythagorean theorem itself.
Using the exercise as an example, with each triangular face having a base of \( 22.0 \) ft (the side of the square base), we would need to know the slant height \( l \) to calculate the area of a single triangle. We can find \( l \) using the Pythagorean theorem, which leads us to the next important concept in geometry: the Pythagorean theorem itself.
Pythagorean Theorem
The Pythagorean theorem is a cornerstone of geometry, applicable in countless scenarios, including the calculation of a pyramid's lateral area. It describes the relationship between the sides of a right-angled triangle: \( a^{2} + b^{2} = c^{2} \), where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides.
In the context of a square pyramid, we can think of one half of the base and the pyramid's height forming a right-angled triangle with the slant height as the hypotenuse. So, for our problem, we calculate the slant height with \[ l = \sqrt{\left(\frac{1}{2} \times 22.0\right)^{2} + 24.5^{2}} \].
This theorem is not only useful for finding distances in geometry problems but also in real-world applications, like navigation and construction.
In the context of a square pyramid, we can think of one half of the base and the pyramid's height forming a right-angled triangle with the slant height as the hypotenuse. So, for our problem, we calculate the slant height with \[ l = \sqrt{\left(\frac{1}{2} \times 22.0\right)^{2} + 24.5^{2}} \].
This theorem is not only useful for finding distances in geometry problems but also in real-world applications, like navigation and construction.
Square Pyramid
A square pyramid, which is the focus of our exercise, has a square as its base and four triangular faces meeting at a single point, the apex. The square pyramid is a special case of a regular pyramid; its properties allow for simpler calculations of volume and surface area.
In our exercise, the pyramid has a base side length of \( 22.0 \) ft and a height of \( 24.5 \) ft. It's crucial to distinguish between the height—perpendicular from the apex to the base—and the slant height, which is the diagonal length along a triangular face from the apex to the midpoint of a base side. Understanding the square pyramid's geometry helps in calculating its volume and lateral area, contributing to better visualisation and practical assessments in architectural designs.
In our exercise, the pyramid has a base side length of \( 22.0 \) ft and a height of \( 24.5 \) ft. It's crucial to distinguish between the height—perpendicular from the apex to the base—and the slant height, which is the diagonal length along a triangular face from the apex to the midpoint of a base side. Understanding the square pyramid's geometry helps in calculating its volume and lateral area, contributing to better visualisation and practical assessments in architectural designs.
Other exercises in this chapter
Problem 29
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