Surface Area and Volume

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Three Dimensional Shapes and Surface Area
CA GR6 AF 3.2, CA GR7 MG 2.1

Since we have learned to find the areas of two-dimensional shapes, we can now apply what we learned to three-dimensional (3-D) objects.

The surface area of a 3-D object is the area of its outside. If we were to wrap a book with gift-wrap perfectly, the surface area of the book would be the area of the gift-wrap.

The surface area for all objects will be the Lateral Surface Area + Base Surface Area.

Prisms

Here are three examples of prisms:

Base surface area is the area of all bases added together. The bases of prisms are the congruent shapes seen on the top and bottom of prisms. Prisms are named after the shape of their bases, as you may see above.

Lateral surface area is the area of all the lateral faces, or the sides of the prism, added together. If you stack fifty triangles on each other, you would get something like the prism on the top right.

The total area of a prism is the sum of the prism's lateral area and the areas of the two bases.

Pyramids

Here are three examples of pyramids:

Notice again how these pyramids are named - by the shape of their base. Pyramids have only one base. The side edges connect to one point. The sides will always be triangles.

Example: Find the lateral area   Find the total surface area

1. Lateral Area = sum of four side areas (bh x 4)
   
   (10 in)(12 in) x 4
   
   = 240 in²
    
2. Total Area = lateral area + one base area
   
   240 in² + (10 in)(10 in)
   
   = 340 in²

Notice that to find the lateral area we needed a value for the line that passes through the middle of each isosceles triangle, as we learned in 2D areas. In these cases, the line is called a slant height.

Circular Solids

These three solids are based on the circle.

A cylinder is like a prism in that it has two parallel and congruent bases. The difference is cylinders have circular bases. If you can remember what a can of beans looks like then you know cylinders already!

The lateral area of the cylinder is the circumference of the cylinder. If you peel the label off of the can of beans, you have the lateral area (see below). Therefore, we would find the circumference of the base circle and multiply it with the height - just like a rectangle.

So then the lateral area of a cylinder must be (C)(h) or 2rh where C is circumference r is the radius and h is the height.

The total area of a cylinder can be found by adding the lateral area with the areas of the two bases:

Cylinder's total area = Lateral area + two base areas (circles).

A cone is like a pyramid however the base is a circle. The slant height is used to find the surface areas of the cone.

The lateral area of a cone is half of the circumference multiplied by the slant height:
Lateral area = (Circumference of the base)(slant height) = rl

The total area of a cone is the lateral area and the base added together.
Total area = Lateral area + area of the circle base = rl + r²

The sphere is special and has no lateral edges and no lateral area. The radius is our only concern in finding the total surface area.

Total surface area of a sphere = 4r²

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