Regular triangular pyramid has the slant height k=12 cm and lateral area AL = 198 cm2. Find the length of the base edge.

To solve this we are going to use the formula for the lateral surface area of a regular pyramid: [tex]LSA= \frac{1}{2} pk[/tex]
where 
[tex]LSA[/tex] is the lateral surface area.
[tex]p[/tex] is the perimeter of the base.
[tex]k[/tex] is the slant height.

We know for our problem that [tex]LSA=198[/tex] and [tex]k=12[/tex], so lets replace those values in our formula and solve for [tex]p[/tex]:
[tex]LSA= \frac{1}{2} pk[/tex]
[tex]198= \frac{1}{2} (p)(12)[/tex]
[tex]198=6p[/tex]
[tex]p= \frac{198}{6} [/tex]
[tex]p=33[/tex]

Now we know that the perimeter of the base of our regular triangular pyramid is 33 cm. Remember that the perimeter of a triangle is the sum of its 3 sides. Since our pyramid is regular, its base will be an equilateral triangle, so to find the length of the base edge, we just need to divide the perimeter by 3:
[tex]base.edge= \frac{33}{3} [/tex]
[tex]base.edge=11[/tex]

We can conclude that the length of the base edge of our regular triangular pyramid is 11 cm.