Find the volume, lateral surface area and total surface area of a regular octagonal pyramid of base 6.2cm and perpendicular height of 14.8cm.

V ≈ 915.7 cm³

LA ≈ 411.3 cm²

SA ≈ 596.9 cm²

Step-by-step explanation:

Volume of a pyramid is:

V = ⅓ Bh

where B is the area of the base and h is the height.

The base is a regular octagon.  The area of a regular octagon is 2(1 + √2) s², where s is the side length.

Substituting:

V = ⅔ (1 + √2) s² h

Given that s = 6.2 and h = 14.8:

V = ⅔ (1 + √2) (6.2)² (14.8)

V ≈ 915.7 cm³

The lateral surface area is the area of the sides of the pyramid.  Each side is a triangular face.  We know the base length of the triangle is 6.2 cm.  To find the area, we first need to use geometry to find the lateral height, or the height of the triangles.

The lateral height and the perpendicular height form a right triangle with the apothem of the octagon.  If we find the apothem, we can use Pythagorean theorem to find the lateral height.

The apothem is two times the area of the octagon divided by its perimeter.

a = 2 [ 2(1 + √2) s² ] / (8s)

a = ½ (1 + √2) s

a ≈ 7.484

Therefore, the lateral height is:

l² = a² + h²

l ≈ 16.58

The lateral surface area is:

LA = 8 (½ s l)

LA = 4 (6.2) (16.58)

LA ≈ 411.3 cm²

The total surface area is the lateral area plus the base area.

SA = 2(1 + √2) s² + LA

SA = 2(1 + √2) (6.2)² + 411.3

SA ≈ 596.9 cm²

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