Question

Architecture The Muttart Conservatory in Edmonton, Alberta consists of four greenhouses in the shape of pyramids. Each of the two largest pyramids has a height of about 24 meters and a base with an area of about 625 square meters. Find the total volume of the two pyramids.

Short answer

The total volume of the two pyramids is 10000 cubic meters.

Step-by-step solution

Recall the formula for the volume of a pyramid

The volume of a pyramid is given by the formula \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \). We will use this formula to find the volume of one of the pyramids.

Calculate the volume of one pyramid

Use the base area \( = 625 \) square meters and height \( = 24 \) meters in the formula. The volume is \( V = \frac{1}{3} \times 625 \times 24 \). Compute \( V = 5000 \) cubic meters.

Calculate the total volume of two pyramids

Since both pyramids have the same dimensions, multiply the volume of one pyramid by 2 to find the total volume: \( 5000 \times 2 = 10000 \) cubic meters.

Key concepts

Geometry Formula

To determine the volume of a pyramid, one must utilize a specific mathematical equation. This equation is relatively simple and highlights the unique properties of pyramidal structures. The formula for calculating the volume of a pyramid is given by \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \).
Let's break it down:

• \( V \) represents the volume of the pyramid, which is what you are trying to calculate.
• The "Base Area" refers to the area of the polygon at the bottom of the pyramid.
• The "Height" is the perpendicular distance from the base to the apex (the topmost point) of the pyramid.
• The fraction \( \frac{1}{3} \) accounts for the tapering shape of the pyramid, which converges at the apex, reducing the overall volume compared to a prism with the same base area and height.

Using this formula ensures that you accurately account for the three-dimensional space a pyramid occupies. Understanding and applying this fundamental geometry formula allows you to effectively solve pyramid volume problems.

Base Area

The base area of a pyramid is a crucial component in determining its volume. In the context of the Muttart Conservatory pyramids, the base area is 625 square meters.
But what exactly is the base area?
Essentially, it's the surface area of the shape that forms the bottom of the pyramid.
Here's a bit more on base areas:

• The base could be any polygon, but in many common pyramids—like the one at the Muttart Conservatory—it tends to be a square or rectangle.
• To find the base area of a rectangle, you multiply the length by the width. For a square, you simply square one of its sides, since all sides are equal.
• The base area affects the stability of the pyramid; larger bases offer better support for taller structures.

Understanding how to calculate the base area is vital, as it directly impacts the calculation of the pyramid's volume. Always ensure you accurately measure or compute the base area when applying it in formulas.

Height of Pyramid

The height of a pyramid, often denoted by \( h \), is an integral part of its volume calculation, as illustrated by the formula \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \).
In the case of our exercise, each pyramid features a height of 24 meters.
Let's delve into what the height of a pyramid entails:

• This height is measured as the perpendicular distance from the center of the base to the pyramid's apex. It is not the slant height, which is the distance along the face of the pyramid.
• The height can significantly influence the overall volume because it determines how tall the pyramid stands.
• If the height is doubled, the volume also doubles, given a constant base area, due to its direct proportionality in the volume formula.

It's crucial to identify the height correctly when solving volume problems. This correct measurement is essential for accurate computations and understanding the pyramid's spatial dimensions.