Question

Can someone help me please? I have the lateral area already but I need to find the total area this regular pyramid.

Answer 1

@tkhunny @Luigi0210

Answer 2

How did you get the lateral area?

Answer 3

LA = 1/2 (perimeter) * (lateral face)

Answer 4

I actually got only partial credit on lateral area. I got\[LA = 18 \sqrt{73}\]

Answer 5

@mathstudent55

Answer 6

What do you mean by "lateral face"?

Answer 7

Let's look at the base of the pyramid.
It's a regular hexagon with side 6.

Answer 8

This is the base of the pyramid.
Each side has length 6, and each angle measures 120 deg.

Answer 9

Now we look at 1/6 of the base.
It's an equilateral triangle with side length 6.

Answer 10

I mean the slant height, sorry! slant height = length of altitude of any lateral face (height of triangular side)

Answer 11

thats what i meant by lateral face ^^

Answer 12

Let's look at one of the 6 triangles.

Answer 13

Since half of the triangle is a 30-60-90 triangle, we can use the ratio of
\(1 : \sqrt 3 : 2\) to find the length of the altitude.

Answer 14

Now we know the altitude, we can find the area of the larger triangle.

\(A = \dfrac{1}{2}bh\)

\(A = \dfrac{1}{2} \cdot 3 \cdot 3 \sqrt 3\)

\(A = \dfrac{9\sqrt 3}{2} \)

Since the large triangle is 1/6 of the entire base, the base (the hexagon) has an area of

\(A = 6 \cdot \dfrac{9\sqrt 3}{2} \)

\(A = \dfrac{54\sqrt 3}{2} = 18\sqrt 3\)

Answer 15

This is the area of the base.
Now we need the lateral area.
The lateral area is the total area of the 6 triangles that go from the base to the point on top.

Answer 16

Since we need to find the area of a triangle, we need a base and an altitude.
The base is 6.
Now we need to find the altitude.