Surface Area and Shapes

Question

A cone has a diameter of 10 in. and a slant height of 13 in. Find the surface area.

A square prism measures 8 km along each edge of the base and is 9 km tall. Find the surface area.

A square pyramid measures 9 yd along the base and has a slant height of 12.8 yd. Find the surface area.

 

 

Please help me find the surface area of these three problems. I already have the formulas however i do not think they are correct so please give me a precise answer and a formula. Thank-you!

 

Answer 1

I'll address the problems as 1, 2, and 3, using the order you listed them in...

 

1.  Thankfully, this one mercifully gives us the highly valued slant height, so we don't have to find it ourselves (not that it would be that difficult if it gave us the radius and height of the cone).  The formula for the surface area of a cone is A = πrs + πr².  The diameter of the cone is given, so in order to find the radius we simply divide d by 2 (10/2 = 5).  Now, we know all of our needed information!

 

r = 5

s = 13

 

A = π*5*13 + π*5² = 65π + 25π = 90π ≈ 282.74 in².

 

2.  For a square prism, the surface area of the entire shape is the area of each of it's faces (or sides) added together.  If it helps you (it helps me) draw it on a piece of paper or picture it in your head.  The prism has two sets of identical sides.  The "top and bottom" sides, which are the squares with sides of 8 km each, and the "side" sides, which are the rectangles that are 9 km tall and 8 km wide.  The area of each of those individual shapes can be found by multiplying length (or height) times the width.

 

For the square faces...

A = 8*8 = 64

 

For the rectangular faces...

A = 9*8 = 72

 

Now, let's add the total number of sides together.  We have a total of six sides, with two being square faces and four being rectangular.

 

64 + 64 + 72 + 72 + 72 + 72 or 2*64 + 4*72 = 416 km²!

 

3.  In the ways as we did for the prism, we can think of the area of the pyramid.  A square pyramid has five faces (one square on the bottom and four triangles with the bases on the edges of the square).  Same as the prism, add up the area of total number of faces to get the surface area of the pyramid.

 

The area of the square is 9*9 = 81.

 

The area of a triangle is 1/2 bh.  b = 9 and h = the slant height = 12.8.  (9*12.8)/2 = 57.6.  

 

Like with the prism, we add the faces together to get the total surface area.

 

81 + 57.6 + 57.6 + 57.6 + 57.6 or 81 + 4(57.6) = 311.4 yd².