To find the lateral surface area, we need to know the perimeter of the base and the slant height. The perimeter is 0.8 × 6 = 4.8, but we need to futz around a bit to find the slant height.
If we split the hexagonal base into six triangles and then bisect them, we'll end up with twelve identical right triangles. The angles facing the center of the hexagon are . That means we have a 30°-60°-90° triangle with the shortest side equaling 0.4 inches (half of the hexagon's side), the hypotenuse equaling 0.8 inches (twice that), and the long side equaling . This might seem pointless, but just go with us.
If we look right at the pyramid instead of down onto it, we see that the altitude and the side (perpendicular to the hexagon edge) create a right triangle with the slant height as the hypotenuse. We can smell the Pythagorean theorem from a mile away.
a2 + b2 = c2
Substitute in our values. The two legs are the altitude (3 inches) and the distance from the center to the side ( inches). The slant height is the hypotenuse.
l2 = 9 + (0.16 × 3) l ≈ 3.08 in2
With the slant height, now we can find the lateral area. We use ½ in the lateral area formula for pyramids because pyramids halve triangles.
That's lateral area. Surface area is just the lateral area plus the area of the base. Since it's a hexagon, the base is going to be a bit tricky. We might not know hexagons that well (nothing against them), but triangles have been with us since the dawn of time. Six triangles have ½ base times height? We know the deal.