Surface Areas

Surface Area is a land that will have you going from plane to plane. The confusion will come when some planes are the exact same as the other. The only way to escape this boldging shaped land is to know the proper formula.

Surface Area Formulas
The surface area is the sum of all the areas of all the shapes that cover the surface of the object.

Cube, Rectangle Prism, Prism, Sphere, Cylinder

Note: "ab" means "a" multiplied by "b". "a²" means "a squared", which is the same as "a" times "a".

Be careful!! Units count. Use the same units for all measurements

Surface Area of a Cube = 6 a ²

the surface area of a cube is the area of the six squares that cover it. The area of one of them is a*a, or a ² . Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.

 Since there are six sides, the total surface area, call it SA is:

SA = a² + a² + a² + a² + a² + a²

SA = 6 × a²

Example #1:


Find the surface area if the length of one side is 3 cm

Surface area = 6 × a²

Surface area = 6 × 3²

Surface area = 6 × 3 × 3

Surface area = 54 cm²


Example #2:


Find the surface area if the length of one side is 5 cm

Surface area = 6 × a²

Surface area = 6 × 4²

Surface area = 6 × 4 × 4

Surface area = 96 cm²


Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac


 

 The surface area of a rectangular prism is the area of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.

The area of the top and bottom (side lengths a and c) = a*c. Since there are two of them, you get 2ac. The front and back have side lengths of b and c. The area of one of them is b*c, and there are two of them, so the surface area of those two is 2bc. The left and right side have side lengths of a and b, so the surface area of one of them is a*b. Again, there are two of them, so their combined surface area is 2ab.

Surface Area of Any Prism

(b is the shape of the ends) Surface Area = Lateral area + Area of two ends (Lateral area) = (perimeter of shape b) * L Surface Area = (perimeter of shape b) * L+ 2*(Area of shape b)

Surface Area of a Sphere = 4 pi r 2

(r is radius of circle)

Surface Area of a Cylinder = 2 pi r 2 + 2 pi r h

(h is the height of the cylinder, r is the radius of the top) Surface Area = Areas of top and bottom +Area of the side Surface Area = 2(Area of top) + (perimeter of top)* height Surface Area = 2(pi r ²) + (2 pi r)* h