Question

A metalworker is creating decorative points in the form of triangular pyramids to go on top of a fence she is building. She wants to know the surface area of each pyramid.

Use the surface area formulas to find the surface area of each pyramid based on the given dimensions:

Base length (L) = 3.9 cm
Base width (W) = 4.5 cm
Height (H) = 3.5 cm

Answer 1

Assuming a right rectangular pyramid (apex above the center of the rectangular base), the surface area is
SA = base area + lateral area = L·W + L·s1 + W·s2,
where s1 = sqrt(H^2 + (W/2)^2) (slant height of the faces with base L) and
s2 = sqrt(H^2 + (L/2)^2) (slant height of the faces with base W).

With L = 3.9 cm, W = 4.5 cm, H = 3.5 cm:

s1 = sqrt(3.5^2 + (4.5/2)^2) = sqrt(12.25 + 5.0625) = sqrt(17.3125) ≈ 4.1608 cm
s2 = sqrt(3.5^2 + (3.9/2)^2) = sqrt(12.25 + 3.8025) = sqrt(16.0525) ≈ 4.0066 cm

Lateral area = L·s1 + W·s2 ≈ 3.9·4.1608 + 4.5·4.0066 ≈ 34.2568 cm^2
Base area = L·W = 3.9·4.5 = 17.55 cm^2

Total surface area ≈ 17.55 + 34.2568 = 51.81 cm^2

So each pyramid has surface area ≈ 51.8 cm^2 (to three significant figures).