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Problem 833

Prove: The lateral area of a regular pyramid is equal to one-half the product of its slant height and the perimeter of I its base.

Short Answer

Expert verified
The lateral area of a regular pyramid can be found by dividing the pyramid into 'n' isosceles triangles, where 'n' is the number of sides of the base. The area of one isosceles triangle is calculated as Area = \( (1/2) * a * l \), where 'a' is the side length and 'l' is the slant height. The total lateral area is the sum of the areas of all isosceles triangles, which can be written as \( A_l = n * (1/2) * a * l \). Replacing \( a * n \) with the perimeter of the base 'P', we get \( A_l = (1/2) * l * P \), which proves that the lateral area of a regular pyramid is equal to one-half the product of its slant height and the perimeter of its base.
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Step 1: Identify key elements and information from the problem description

A regular pyramid has a polygonal base with equal sides and isosceles triangular faces that meet at the peak of the pyramid, called the apex. Let's denote the base's side length as 'a', the number of sides of the base as 'n', the slant height as 'l', and the lateral area as 'A_l'. Note that the lateral area does not include the area of the base.

Step 2: Divide the pyramid into isosceles triangles

In order to find the lateral area, we want to find the sum of the areas of all the isosceles triangular faces. To do this, we'll divide the pyramid into 'n' isosceles triangles, where 'n' is the number of sides of the base.

Step 3: Calculate the area of one isosceles triangular face

To find the area of one isosceles triangle, we will use the formula for the area of a triangle: Area = (1/2) * base * height. In the case of one isosceles triangle of the pyramid, the base of the triangle corresponds to the side length 'a' and the height of the triangle corresponds to the slant height, 'l'. So the area of one isosceles triangle is: Area = (1/2) * a * l.

Step 4: Determine the perimeter of the base

Since all sides of the base are equal in length, the perimeter of the base is equal to the product of the side length 'a' and the number of sides 'n'. So, the perimeter of the base is P = a * n.

Step 5: Multiply the area of one isosceles triangular face by the number of triangular faces

We'll now use the area of one isosceles triangle that we found in Step 3, and multiply it by the number of triangular faces 'n'. This will give us the total lateral area of the pyramid: A_l = n * (1/2) * a * l.

Step 6: Prove the given formula for the lateral area of a regular pyramid

We can now use the fact that P = a * n, the perimeter of the base, to connect our equation with the given formula for the lateral area of the pyramid. By substituting P for a * n in our equation, we get A_l = (1/2) * l * P, which proves that the lateral area of a regular pyramid is equal to one-half the product of its slant height and the perimeter of its base.

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