Applied Math: Three-Dimensional Shapes

Solids
The surface area of a solid object is the area of all sides or exterior surfaces.

For objects such as prisms and pyramids, a further distinction is made between base surface area (B) and lateral surface area (LA).

For a prism, the total surface area (SA) is .

For a pyramid or cone, the total surface area is .
The surface area of a sphere can be found by the formula , where r is the radius.

The volume is given by the formula , where r is the radius.

Both quantities are generally given in terms of π.

The volume of any prism is found by the formula , where B is the area of the base, and h is the height (perpendicular distance between the bases).

The surface area of any prism is the sum of the areas of both bases and all sides. It can be calculated as , where P is the perimeter of the base.

For a rectangular prism, the volume can be found by the formula , where V is the volume, l is the length, w is the width, and h is the height.

The surface area can be calculated as  or .

The volume of a cube can be found by the formula , where s is the length of a side.

The surface area of a cube is calculated as , where SA is the total surface area and s is the length of a side.

These formulas are the same as the ones used for the volume and surface area of a rectangular prism, but simplified since all three quantities (length, width, and height) are the same.
The volume of a cylinder can be calculated by the formula , where r is the radius, and h is the height.

The surface area of a cylinder can be found by the formula .

The first term is the base area multiplied by two, and the second term is the perimeter of the base multiplied by the height.

The volume of a cone is found by the formula , where r is the radius, and h is the height.

Notice this is the same as  times the volume of a cylinder.

The surface area can be calculated as , where s is the slant height.

The slant height can be calculated using the Pythagorean theorem to be , so the surface area formula can also be written as .


Practice
P1. Find the surface area and volume of the following solids:
(a) A cylinder with radius 5 m and height 0.5 m.
(b) A half sphere (radius 5 yds) on the base of an inverted cone with the same radius and a height of 7 yds.

Practice Solutions:
P1.

(a) ;

(b) We can find s, the slant height using the Pythagorean theorem, and since this solid is made of parts of simple solids, we can combine the formulas to find surface area and volume: