Question

If a square based pyramind has sides of 20 what is the the height. All sides are 20 and the triangles are obviously, equilatteral. Thanks :)

Answer 1

Take one of the equilateral triangles and drop a perpendicular from the top to the base. The perpendicular bisects the base and forms two back-to-back right-angled triangles. The length of the perpendicular is sqrt(20^2-10^2) by Pythagoras. That's sqrt(300)=10sqrt(3). Now view the pyramid side on and drop a perpendicular from the apex where the four equilateral triangles meet on to the square base. The perpendicular meets the base at the centre of the square. Take just one of the triangular sides and view it joining the perpendicular from the apex. We have a right-angled triangle where the hypotenuse has length 10sqrt(3) and the base is half the side of the square so its length is 10. The third side of this internal triangle is the height of the pyramid=sqrt(300-10^2) where 300 is (10sqrt(3))^2. So the height is sqrt(200)=10sqrt(2)=14.14 approx.