To find the slant height of a triangular pyramid, we first need to find the height of one of the faces of the pyramid. Since the pyramid is made up of equilateral triangles, we can use the formula for the area of an equilateral triangle to find the height.
The formula for the area of an equilateral triangle is A = (sqrt(3)/4) * s^2, where s is the length of one side of the equilateral triangle.
Given that the total surface area of the triangular pyramid is 78 square inches, we can find the side length of one of the equilateral triangles by dividing the surface area by the number of faces on the pyramid. Since there are 4 equilateral triangles in a triangular pyramid, this will give us the area of one equilateral triangle.
78 in^2 / 4 = 19.5 in^2
Now we can find the side length, s, by taking the square root of this area:
s = sqrt(19.5) = 4.416 in
Now that we have the side length of the equilateral triangle, we can find the height of the triangle using the formula:
height = (sqrt(3)/2) * s
height = (sqrt(3)/2) * 4.416
height = 7.64 in
Finally, to find the slant height, we can use the Pythagorean theorem, where the slant height is the hypotenuse of a right triangle formed by the height and the side length of the equilateral triangle.
Slant height = sqrt( height^2 + (s/2)^2)
Slant height = sqrt( 7.64^2 + (4.416/2)^2)
Slant height = sqrt(58.23 + 2.208)
Slant height = sqrt(60.438)
Slant height = 7.78 inches
Therefore, the slant height of the triangular pyramid is approximately 7.78 inches.
Use the net to find the slant height of the triangular pyramid made of equilateral triangles if its surface area is 78 in.^2. ( 6 in. 5 in. )
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