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The surface area of the pyramid is superfluous to calculating the slant height as the slant height is the height of the triangular side of the pyramid which can be worked out using Pythagoras on the side lengths of the equilateral triangle:

side² = height² + (½side)²

→ height² = side² - ¼side²

→ height² = (1 - ¼)side²

→ height² = ¾side²

→ height = (√3)/2 side

→ slant height = (√3)/2 × 9cm = 4.5 × √3 cm ≈ 7.8 cm.

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However, the surface area can be used as a check:

140.4 cm² ÷ (½ × 9 cm × 7.8 cm) = 140.4 cm² ÷ 35.1 cm² = 4

So the pyramid comprises 4 equilateral triangles - one for the base and 3 for the sides; it is a tetrahedron.

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Q: What is the slant height of a pyramid that has all sides as equilateral triangles with sides of length of 9 cm and the surface area of the pyramid is 140.4 square cm?

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