Question

Given: MPNA pyramid All edges congruent AO = 36, AO ⊥ (MNP) Find: m∠AMO, AΔANM

Answer 1

Answer:

m∠AMO ≈ 54.7°

AΔANM = 486√3

Step-by-step explanation:

The edges are congruent, so all four faces are congruent equilateral triangles.  We'll say the length of each edge is 2r.

The height of the pyramid h is given to be 36.

The perpendicular distance from O to line MP is called the apothem (a).  Using 30-60-90 triangles, b = 2a and r = a√3.

Use cosine to find m∠AMO.

cos(∠AMO) = b / (2r)

cos(∠AMO) = (2a) / (2a√3)

cos(∠AMO) = 1 / √3

m∠AMO ≈ 54.7°

Use Pythagorean theorem to find the apothem.

(2r)² = b² + h²

(2a√3)² = (2a)² + 36²

12a² = 4a² + 1296

8a² = 1296

a² = 162

a = 9√2

So the edge length is:

2r = 2√3 (9√2)

2r = 18√6

The area of the equilateral triangle ΔANM is half the apothem times the perimeter:

A = ½aP

A = ½ (9√2) (3 × 18√6)

A = 243√12

A = 486√3