Understanding Surface Area
Understanding surface area is essential for measuring the total area that an object occupies, which has various practical applications, such as determining how much material is needed to cover an object or predicting sound transmission. Surface area is generally expressed in square meters (m²) and can be calculated for various shapes, including polyhedrons—three-dimensional figures with flat faces. For polyhedrons, the total surface area is the sum of the areas of each face, while for curved surfaces, the calculation is more complex due to their infinite small faces.
Composite figures, formed by combining simpler shapes, also require careful calculation of surface area, particularly where the shapes meet. For cylinders, the surface area can be easily calculated based on the area of a circle and the height of the cylinder, while cones and pyramids use different formulas that consider their unique dimensions and shapes. Lastly, the surface area of spheres presents an interesting relationship, being four times that of a circle with the same radius, revealing deeper mathematical properties. This comprehensive understanding of surface area across different shapes underscores its importance in both theoretical and practical contexts.
Understanding Surface Area
The surface area of an object is a measurement of the total area that the object occupies. The surface area of an object has many useful interpretations—for instance, the surface area of an object describes exactly how much material to use if you need to cover an object, or the surface area of an object that will be exposed to sound can help you predict how the sound will pass through it.
In standard units, surface area is measured in square meters (m2). Polyhedrons are objects with flat faces, in that their edges connecting to different vertices are straight. Finding the total surface area of a polyhedron is the sum of the surface area for each face. For curved surfaces, the mathematical definition for surface area is much more complex than that for a simple polyhedron, because a curved object can be broken into infinitely small faces, each of which have a different surface area. Nonetheless, the same approach is taken—except the surface is mathematically represented, and its surface area is summed over each part.
Surface Area of a Composite Figure
Composite figures are the composition, or combination, of other, smaller figures. Because surface area can be broken up by finding the area of each of the smaller faces, then if the surface is a combination of several objects whose surface area may be calculated, the total surface area is the sum of the surface area of each of the composite figures (except where the faces on the adjacent figures meet).
For instance, the shape of a simple house can be viewed as a rectangular prism and a triangular prism as the roof. The surface area of the entire house will be the sum of the surface area of the entire rectangular prism and the triangular prism, minus the face where the prisms meet (because it is not part of the surface of the new, composed figure).
Often students make the mistake of forgetting to account for cases in which the faces of two objects with known surface area meet. A clever approach to avoid this is to represent the figure as a two dimensional object called a "net," which describes all of the faces on the outer surface in two-dimensional flat space. By then calculating the area of each of these figures and adding them together, the surface area of the entire object can be determined.
Surface Area of a Cylinder
In mathematics, a circle and a cylinder have the same definition, except a cylinder is the set of points that are also projected along a third dimension, rather than just two dimensions. Therefore, the surface area of a cylinder is very easy to calculate. Since the area of a circle is πr2, and if a cylinder is projected h units along the z-axis, the outside surface area of the cylinder is simply πr2h.
This is called the lateral surface area because it does not include the top and bottom of the cylinder. Since the top and bottom of a cylinder are two congruent circles, you can add their areas to the lateral surface area of the cylinder and obtain its total surface area.
Surface Area of a Cone
A right circular cone is an object that tapers smoothly from a circular base to a point called the vertex. The entire surface area of a cone may be expressed as the sum of the base and its lateral surface area. The lateral surface area of the one is given by πrs, where s is the slant height of the cone and r is the radius of the circular base.
If you only have the height of the cone, you can express s using the Pythagorean theorem, where s is the hypotenuse of a right triangle formed by the radius of the circular base and the height of the cone. Thus the formula for the lateral surface area in terms of the height is:
The outside surface area is simply that of a circle; therefore the total surface area of the right circular cone is:
Surface Area of a Pyramid
There are many different types of pyramids in mathematics, the most common being a right square–based pyramid. A right square–based pyramid is one in which a vertical line can be drawn from its vertex that touches the midpoint of its square base.
The surface area of a right square–based pyramid is as follows:
where b is the side length of the square base, and s is the slant height.
Since the pyramid is a right pyramid, the slant height may also be expressed in terms of the height using the Pythagorean theorem. Thus the surface area of the pyramid in terms of the height becomes:
Other forms of pyramids will have different surface areas. A very common example is a tetrahedron. A tetrahedron is a right pyramid as well, except it is composed of four equilateral triangles, one being the base of the pyramid.
Since the equilateral triangles are congruent, their surface area is four times the surface area of one of the equilateral triangles, or simply:
where s is the slant height of the pyramid and l is the side length of one of the faces. By expressing the slant height in terms of the side length, this can be rewritten as:
Surface Area of a Sphere
The surface area of a sphere with radius r is
. Interestingly, this is exactly four times the area of a circle with the same radius. It is also one of few surfaces where the rate of change of its volume as you vary the radius is exactly equal to its surface area. Archimedes concluded that this was due to the fact that the volume of a sphere can be viewed as the surface area of an infinite number of concentric circles stacked on each other, whose perimeter forms the surface area of the entire sphere. It is a property that many symmetrical objects have, but the elegance of its simplicity is astonishing.
Bibliography
Bass, Alan. Geometry: Fundamental Concepts and Applications. Upper Saddle River, NJ: Pearson, 2007. Print.
Byer, Owen, Felix Lazebnik, and Deirdre L. Smeltzer. Methods for Euclidean Geometry. Washington, DC: Mathematical Association of America, 2010. Print.
Kaur, Berinderjeet. Mathematical Applications and Modelling: Yearbook 2010. Hackensack: World Scientific, 2010.
McKellar, Danica. Girls Get Curves. New York: Hudson Street, 2012.
Merzbach, Uta C., and Carl B. Boyer. A History of Mathematics. 3rd ed. Hoboken: Wiley, 2011.
Stankowski, James F., ed. Geometry and Trigonometry. New York: Rosen, 2015.