Surface Area

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Surface Area


Please review Measurements first.

Surface area usually refers to three dimensional objects, however one sometimes talks about the surface area of a flat surface such as a table top, a wall, a field, that is, just one side of the object.  However the term "surface area" refers to the measure of the "amount of surface" enclosing some object, or the amount of surface within some object.  At the end I provide ways to visualize these shapes.

Consider these shapes:


Each face has an area, and each face is in a pair that have equal areas.  The front and back of this box have an area equal to LH, so the area of the front and the back equals  2LH.  Likewise the top and bottom forms a pair each with an area of LW;  they contribute 2LW, and finally the left and right ends contribute an area of 2WH.  We have no other surfaces on the exterior of this box, therefore the total surface area is 2LH + 2LW + 2WH = 2(LY + LW + WH).

If we have a cube then all dimensions are equal, that is, L=W=H.  In this case let s equal this dimension and The surface are of a cube would be

 6( ss ) = 6s2.      (6 sides to the square each with area s2.)


The surface area of a tennis ball is about (4)(3.14)(1.5in)2 = 28 square inches.  The surface area of a #10 soccer ball is about (4)(3.14)(4in)2 = 201 square inches.  So, the soccer ball is about 7 times larger than the tennis ball as far as surface area is concerned.

These numbers man seem large, but recall that we are dealing with a curved surface, not a flat on, when you place a square inch on the surface the corners are pulled in; imagine  placing a stamp on a tennis ball, the stamp gets deformed because of the curvature of the ball.



A right circular cylinder is a cylinder that has circular ends and the wall is perpendicular to both of these ends  Now the surface area of this cylinder is comprised of the areas of the ends (circles) and  the area of the wall. The wall is a rectangle with the width equal to h  and the length equal to the circumference of either end.  The are of a circle is  πr2  and its circumference is 2πr;  se we get  2(πr2) +  (2πr)h as the surface area.  The 2, the π, and the r can be factored out to arrive at the equation above.



A right circular cone is a cone that has a circular base and  whose altitude (height) from circle center to the apex of the cone is perpendicular to the base.  In the figure above the radius r is perpendicular to the height h.  For example if the radius equals 10 cm, and the height is 25 cm, then the surface area is

(25cm)2 + (10 cm)2 = 625cm2 + 100cm2 = 725cm2; taking the square root we get s = 26.9 cm

so the surface area would be 2(3.14)(10cm)(26.9cm)  + (3.14)(10cm)2 = 1690.9 cm+ 314 cm2 = 2005 cm2


Here are ways to visualize the surface areas above and are great ways to provide a hands on experience for your student.


First the box.   Find a cardboard box and label the sides.  Cut it then unfold it until it looks like the unfolded image below.  Or, start with the unfolded image, have your student create that image with the dimensions you provide, find the total area, then fold it back into place to create the box. 



Below is a right circular cylinder that is being rolled along a sheet of paper whose width is the height of the cylinder.  The numbers are provided to show the positions along the circumference of an end to keep track of when to stop rolling.  As you can see the rectangular sheet of paper has a length equal to the circumference of the one end of the cylinder.




Finally, shown is a cone rolled along a sheet of paper with its point remaining in one place.  One complete revolution of the cone traces out a sector of a larger circle on the paper.  The area of this sector is the area of the curved surface of the cone.  To carry out this exercise you must make sure the point of the cone does not change position as the cone rotates around the paper.








This is another FREE ALGEBRA PRINTABLE presented to you from the Algebra section of


Download our free math lesson plan template...and print!!

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