A pyramid is a polyhedron in which one face (the base) can be any polygon, and the other faces (the lateral faces) are triangles that meet at a common vertex called the vertex of the pyramid. Similar to an altitude of a triangle, the altitude of a pyramid is a segment from the vertex that is perpendicular to the plane of the base.
The length of the altitude is called the height of a pyramid. A pyramid is regular if its base is a regular polygon and its lateral faces are congruent, isosceles triangles. In a regular pyramid, the length of the altitude of each lateral face is sometimes referred to as the slant height of a pyramid.
If the vertex of a pyramid is not over the midpoint of its base, it is called an oblique pyramid.
When the base area and the height of a pyramid are known, its volume can be calculated.
Consider a pyramid and a prism that have the same base area and height.
A pyramid can be modeled as a stack of prisms. The sum of the volumes of the small prisms will be greater than the pyramid's volume. However, as the number of prisms increases and they get thinner, the sum will approximate the volume of the pyramid.
Furthermore, the ratio of the sum of the volumes of each small prism to the volume of the prism will be approximated to
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Therefore, the volume of a pyramid is one third of the prism with the same base area and height.
Start by finding the area of the base of the tent. Then, the formula for the volume of a pyramid will be used.
The base of the tent is a regular pentagon with side lengths meters. Therefore, its perimeter is times Now, draw the apothem of the pentagonal base. The measure of each central angle of a regular pentagon is and is bisected by the apothem.
Dominika goes to watch a basketball match at the Walter Pyramid in Long Beach, California. She is amazed by the appearance of the arena. She finds out that the arena was built on a square base with side lengths of feet.
Architects might enjoy turning things upside down. Maya is interested in architecture and follows some online magazines about it. After reading an article about the Slovak Radio Building in Bratislava, Slovakia, she wonders about the area of the square rooftop.
If the height of the building is meters and its volume is about cubic meters, find the area of the rooftop of the building.
Designers and inventors also benefit from pyramids. An object attracts Mark's attention on a school trip to a maritime museum. The guide explains that it is called a deck prism, which was invented to illuminate the cabins below deck before electric lighting. Mark buys a replica of the deck prism, which is composed of a prism and pyramid, each with a regular hexagonal base.
The deck is composed of two solids:
Therefore, the volume of the deck is the sum of the volumes of the above solids. The volume of each solid will be found one at a time.
The surface area of a regular pyramid can be calculated using the following formula.
In this formula, is the perimeter of the base, is the base area, and is the slant height. In the case that the pyramid is not regular, the area of each lateral face has to be calculated one by one and then added to the area of the base.
A regular pyramid's surface area can be seen as two separate parts: the lateral area and the base. Since for a regular pyramid, its base can be any sided regular polygon, the lateral area is the sum of the area of congruent triangles. For example, consider a regular hexagonal pyramid with an edge length and a slant height Take a look at its net.
As can be seen, the area of each lateral face is Therefore, the lateral area will be times because the lateral faces are congruent. Notice that is the perimeter of the base, which can be denoted by Then, the lateral area can be expressed as follows. Therefore, the formula for the surface area is obtained.
Maya's father decides to cover the roof of their house with waterproof insulation material. Maya's father asks Maya to calculate how many square feet of insulation material is needed.
The roof is a square pyramid with a height of feet and base side length of feet. Help Maya calculate the area.