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Chapter 12: Problem 1

Explain the difference between the slant height and the altitude of a regular pyramid.

Short Answer

Expert verified
The altitude is vertical and perpendicular to the base, while the slant height is an inclined line from apex to base edge midpoint.

Step by step solution

01

Define Regular Pyramid

A regular pyramid is a three-dimensional geometric figure with a polygonal base and triangular faces that converge at a single point called the apex. The base of the pyramid is regular, meaning all its sides are equal, and it's generally referred to as having a specific height.
02

Describe the Altitude of a Pyramid

The altitude of a regular pyramid is the perpendicular distance from the apex (top point) of the pyramid to the center of the base. It is a vertical line and is also known as the height of the pyramid.
03

Describe the Slant Height of a Pyramid

The slant height is the distance from the apex of the pyramid to the midpoint of one of the edges of the base face. In a regular pyramid, it is not vertical but inclined, forming a slant with respect to the base of the pyramid.
04

Compare Altitude and Slant Height

The key difference between the two is their orientation: the altitude is a vertical line from the apex perpendicular to the base, while the slant height is an inclined line from the apex to the midpoint of a base edge. They are used to calculate different properties of the pyramid, such as volume and surface area.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Regular Pyramid
A regular pyramid is a fascinating 3D shape. It has a base that's a regular polygon, which means all sides and angles are equal. Imagine a pyramid like the famous ones in Egypt but with a perfectly symmetrical base. From each vertex of the base, a triangular face rises to meet at a point above the base called the apex. This gives the pyramid its distinct shape.
  • All lateral faces of a regular pyramid are congruent, meaning they have the same size and shape.
  • The apex and the center of the base are directly aligned vertically in a regular pyramid.
  • The symmetry in a regular pyramid makes calculation easier, especially when finding volume and surface area.
Understanding regular pyramids is crucial because it forms the basis for differentiating key measurements like the slant height and altitude.
Slant Height
The slant height in a regular pyramid is quite unique. It's not the same as the vertical height, and this can sometimes cause confusion. Picture the slant height as the side of the triangle that forms between the apex and the edge of the base.
It's called 'slant' because it's inclined rather than vertical. The slant height stretches from the apex down to the midpoint of any side of the base.
  • In mathematical terms, if you imagine opening up a pyramid into its net, the slant height is the height of each triangular face of the pyramid.
  • This measurement is essential when calculating the lateral surface area of the pyramid, as it directly affects the area of each triangular face.
Many problems involving pyramids will specify either the slant height or require it to be found to solve for other dimensions or attributes.
Altitude
Altitude in geometry, specifically for pyramids, is a critical concept to grasp. This is the height that measures the perpendicular distance from the very top of the pyramid, the apex, straight down to the center of the base.
It's an absolute vertical measure, unlike the slant height.
  • You can think of the altitude as the spine of the pyramid, being purely upright and creating a perfect line down to the base.
  • The altitude is fundamental when you calculate the volume of the pyramid, as it represents the true height needed in the volume formula: \( V = \frac{1}{3} \times \text{Base Area} \times \text{Altitude} \).
The clarity of this concept helps in distinguishing it from other measures like the slant height and is therefore vital in understanding the true scale and proportion of the pyramid.

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Most popular questions from this chapter

Short Response Find the area of a regular octagon whose perimeter is \(49.6\) feet and whose apothem is \(7.5\) feet long.

Draw an example of each solid. a. cylinder b. pentagonal prism c. cone d. rectangular pyramid

Miniatures The Carole \& Barry Kaye Museum of Miniatures in Los Angeles displays a tiny desk made from a thousand pieces of wood. a. If the 3-inch tall desk represents a real desk that is 30 inches tall, what is the scale factor of the miniature to the real desk? b. What is the ratio of the surface areas of the miniature to the real desk? c. What is the ratio of the volumes of the miniature to the real desk?

History The Great Pyramid at Giza was built about 2500 B.C. It is a square pyramid. a. Originally the Great Pyramid was about 481 feet tall. Each side of the base was about 755 feet long. What was the original volume of the pyramid? b. Today the Great Pyramid is about 450 feet tall. Each side of the base is still about 755 feet long. What is the current volume of the pyramid? c. What is the difference between the volume of the original pyramid and the current pyramid? d. What was the yearly average (mean) loss in the volume of the pyramid from 2500 B.C. to 2000 A.D.?

Find the area of each figure to nearest hundredth. circle: diameter, 7 meters
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