Question

a triangular pyramid with an equilateral base has a side length of 10 centimeters and a surface area of 214.5 square centimeters. find its slant height

Answer 1

We know that
[surface area of a triangular pyramid]=[area of the base]+3*[area of the lateral triangle]

step 1
find the area of the base
is a equilateral triangle
applying the Pythagorean theorem

h²=10²-5²------> h²=100-25----------> h=√75 cm
[area of the base]=10*√75/2---------> 5√75 cm²

step 2
find the area of the lateral triangle
[surface area]=[area of the base]+3*[area of the lateral triangle]
[area of the lateral triangle]={[surface area]-[area of the base]}/3
[area of the lateral triangle]={[214.5]-[5√75]}/3
[area of the lateral triangle]=57.07 cm²

step3
find the slant height
[area of the lateral triangle]=b*[slant height]/2
[slant height]=2*[area of the lateral triangle]/b
[slant height]=2*[57.07]/10-----> [slant height]=11.41 cm

the answer is 
the slant height is 11.41 cm