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Chapter 48: Chapter 48

Problem 830

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Find the surface area of a regular tetrahedron when each edge is of length a) \(1 ;\) b) 2 .

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a) The surface area of a regular tetrahedron with edge length 1 is \( \sqrt{3} \) square units. b) The surface area of a regular tetrahedron with edge length 2 is \( 4\sqrt{3} \) square units.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Recall the formula for the area of an equilateral triangle

For an equilateral triangle with side length \(s\), the area (A) can be computed using the formula: \(A = \frac{\sqrt{3}}{4}s^2\).

Step 2: Calculate the area of one equilateral triangular face for each case

a) When the side length (\(a\)) is 1, the area of one equilateral triangle is: \(A_1 = \frac{\sqrt{3}}{4}(1)^2 = \frac{\sqrt{3}}{4}\) b) When the side length (\(a\)) is 2, the area of one equilateral triangle is: \(A_2 = \frac{\sqrt{3}}{4}(2)^2 = \frac{4\sqrt{3}}{4} = \sqrt{3}\)

Step 3: Find the surface area of the regular tetrahedron for each case

Since there are four equilateral triangular faces in a tetrahedron, we need to multiply the area of one face by 4. a) Surface area of the tetrahedron with side length 1: \(SA_1 = 4A_1 = 4\left(\frac{\sqrt{3}}{4}\right) = \sqrt{3}\) b) Surface area of the tetrahedron with side length 2: \(SA_2 = 4A_2 = 4(\sqrt{3}) = 4\sqrt{3}\)

Step 4: Results

a) The surface area of a regular tetrahedron with edge length 1 is \(\sqrt{3}\) square units. b) The surface area of a regular tetrahedron with edge length 2 is \(4\sqrt{3}\) square units.

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