Mathematics: Geometry - 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
is the radius. The volume is given by the formula <br><img src=" />r is the radius. Both quantities are generally given in terms of π.


The volume of any prism is found by the formula
is the area of the base, and <i>h</i> 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 <br><img data-cke-saved-src=" />, where P is the perimeter of the base.

For a rectangular prism, the volume can be found by the formula
is the volume, <i>l</i> is the length, <i>w</i> is the width, and <i>h</i> is the height. The surface area can be calculated as <br><img data-cke-saved-src=" />
or
.

The volume of a cube can be found by the formula
is the length of a side. The surface area of a cube is calculated as <br><img src=" />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
is the radius, and <i>h</i> is the height. The surface area of a cylinder can be found by the formula <br><img data-cke-saved-src=" />. 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 pyramid is found by the formula
from the vertex to the base). Notice this formula is the same as <br><img data-cke-saved-src=" />
times the volume of a prism. Like a prism, the base of a pyramid can be any shape.
Finding the surface area of a pyramid is not as simple as the other shapes we've looked at thus far.
If the pyramid is a right pyramid, meaning the base is a regular polygon and the vertex is directly over the center of that polygon, the surface area can be calculated as
is the perimeter of the base, and <i>h_s</i> is the slant height<br> (distance from the vertex to the midpoint of one side of the base). If the pyramid is irregular, the area of each triangle side must be calculated individually and then summed, along with the base.<br><img data-cke-saved-src=" />

The volume of a cone is found by the formula
is the radius, and <i>h</i> is the height. Notice this is the same as <br><img data-cke-saved-src=" />
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
.

 

P1. Find the surface area and volume of the following solids:
(a) A cylinder with radius 5 m and height 0.5 m. trapezoidal prism with base area of 254
, base perimeter 74 mm, and height 10 mm.
(c) A half sphere (radius 5 yds) on the base of an inverted cone with the same radius and a height of 7 yds.
P1. (a)
;


(b)
;


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