A square based pyramid has a base with an area of 9m squared and a height of 4 m find the surface area

Question

It's a geome qu 

Answer 1

Hi Yara,

 

Each triangular face of your pyramid has the same area.  So if we can figure out the area of one triangular face, we will multiply it by 4, then add the area of the base, and that will give us the surface area.

 

The base is a square with an area of 9 m^2.  This means that each side of the square is 3 m long.  (A of a square is (base)(height) and in a square, these are the same, so A = (side length)^2, so side length = sqrt(9) = 3 m.

 

This 3 m side of the square is the base of one of the triangles that make up the four faces.  If the height of the pyramid (the perpendicular distance from the base to the vertex at the top) is 4, we can make a right triangle inside the pyramid to determine the height of each triangular face (the slant height).

   

The vertical edge of the triangle is the height = 4 m.  

The horizontal edge of the interior right triangle will be 1/2 of the side length, so the base = 1.5 m

The slant height (the height of each triangular face) is found using the Pythag Thm:  

 

4^2 + 1.5^2 = (slant height)^2.

 

So (slant height) =sqrt(18.25) =4.272

 

So the height of each triangular face is 4.272 m.

 

Each triangular face then has an area of (1/2)(3)(4.272) =6.408

 

4 faces = 4 * 6.408 = 25.632

1 base = 9 

 

Total surface area is 9 + 25.632

 

= 34.9632 square meters.