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Problem 830

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

Expert verified
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.
See the step by step solution

## 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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