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Magazine Chapter 5: Problem 209
(a) Find a formula for the surface area of a right circular cone with base of radius \(r\) and slant height \(l\). (b) Find a similar formula for the surface area of a right pyramid with apex \(A\) whose base \(B C D E \cdots\) is a regular \(n\) -gon with inradius \(r\).
Short Answer
Expert verified
The surface area of the cone is \( \pi r (r + l) \) and for the pyramid is \( \frac{1}{2} n s (r + l) \).
Step by step solution
01
Understanding the Cone Components
A right circular cone consists of two parts: the circular base and the lateral surface. The base has a radius of \( r \), and the slant height of the cone is given as \( l \).
02
Calculating the Base Area of the Cone
The area of the circular base can be calculated using the formula for the area of a circle: \( A_{\text{base}} = \pi r^2 \).
03
Calculating the Lateral Surface Area of the Cone
The lateral surface area of the cone is a sector of a circle with the radius as the slant height \( l \) and a circumference of \( 2\pi r \). The formula for the lateral surface area of the cone is \( A_{\text{lateral}} = \pi r l \).
04
Summing the Areas for Total Cone Surface Area
The total surface area of the cone is the sum of the base area and the lateral surface area. Therefore, \( A_{\text{total}} = \pi r^2 + \pi r l = \pi r (r + l) \).
05
Understanding the Pyramid Components
A right pyramid with an apex \( A \) and a regular \( n \)-gon base consists of the base and the lateral triangular faces. The base is a regular \( n \)-gon with inradius \( r \).
06
Calculating the Base Area of the Pyramid
The area of the regular \( n \)-gon base with inradius \( r \) and side length \( s \) can be calculated using the formula for the area of a regular polygon: \( A_{\text{base}} = \frac{1}{2} n s r \).
07
Calculating the Lateral Surface Area of the Pyramid
Each triangular face has an area given by \( \frac{1}{2} s l \), where \( s \) is the side length and \( l \) is the slant height. The lateral surface area of the pyramid is \( n \times \frac{1}{2} s l = \frac{1}{2} n s l \).
08
Summing the Areas for Total Pyramid Surface Area
The total surface area of the pyramid is the sum of the base area and the lateral surface area: \( A_{\text{total}} = \frac{1}{2} n s r + \frac{1}{2} n s l = \frac{1}{2} n s (r + l) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Surface Area
Surface area is a key concept in geometry, describing the total space that covers a three-dimensional shape. It consists of adding up the areas of all the surfaces of a solid object. For 3D shapes, it's essentially the outer layer or shell you can physically touch.
When calculating the surface area, each part of the shape - like the base, the sides, or any curved parts - contributes to the final answer.
When calculating the surface area, each part of the shape - like the base, the sides, or any curved parts - contributes to the final answer.
- For shapes like a cone, you'll consider both the circular base and the side known as the lateral surface.
- For a pyramid, you need to account for both the base, that is often a different polygon, and the triangular faces that make the sides.
Right Circular Cone
A right circular cone is a 3-dimensional geometric shape, having a circular base and a single apex point. The apex of a right circular cone is directly above the center of its base.
Understanding its components is essential:
Understanding its components is essential:
- Base: A flat circular surface with a specific radius denoted as \( r \).
- Slant Height (\( l \)): The distance measured along the cone's surface, from the base's edge directly up to the apex.
- Base Area: Calculated as \( \pi r^2 \).
- Lateral Surface Area: This is the side part of the cone, which is calculated with the formula \( \pi r l \).
Right Pyramid
A right pyramid is characterized by having a polygonal base and triangular faces that converge at a single apex directly above the center of the base. If the base is a regular polygon, the pyramid is known as a regular pyramid.
Let's break down its components:
Let's break down its components:
- Base: This is a regular \( n \)-gon, meaning all sides and angles are equal.
- Apex: The topmost point of the pyramid, positioned such that it is directly above the center of the base.
- Inradius (\( r \)): The distance from the center of the base to the midpoint of a side of the \( n \)-gon.
- Base Area: Determined using \( \frac{1}{2} n s r \), where \( s \) is the side length.
- Lateral Surface Area: This consists of triangular faces, calculated as \( \frac{1}{2} n s l \), with \( l \) being the slant height.
Regular n-gon
A regular \( n \)-gon is a polygon with \( n \) sides, where all the sides and angles are equal. This geometric figure provides a symmetric and visually pleasing shape, commonly used as the base of a pyramid.
Key features of a regular \( n \)-gon:
Key features of a regular \( n \)-gon:
- Equal Sides: Each side has the same length, denoted as \( s \).
- Central Angles: Each interior angle and each central angle are always the same across the shape.
- Inradius: The perpendicular distance from the center of the \( n \)-gon to the midpoint of any side, typically denoted as \( r \).
