# Surface Area

 Surface area is a two-dimensional measure of a three-dimensional geometric figure. It measures the outside surface, or the combined areas of each face. It can be found by decomposing the figure into various flat pieces, then we can then easily figure out the area of each. Surface area is always given in square units. In the picture above, the first green figure is a cube with side measures of 2 units. There are six faces, each with area 2 X 2 = 4, so the surface area is 6 X 4 = 24 units squared. The second figure is a purple "brick" or rectangular prism, with measures 1 X 1 X 4. We will need to find the area of each face. The bottom, top, front and back faces are the same, with area 1 X 4 = 4 square units each. Four of these add up to 16 square units. Then the end pieces are squares with area 1 X 1 = 1 square unit each, and both add up to 2 square units. Adding up the areas of all the faces gives us 16 + 2 = 18 square units. The blue cylinder is made up of two circles and a rectangle (think of the label on a soup can). We can find the area of each circle of radius two by multiplying (2)2(pi) = 12.56 square units. The rectangle "label" is three inches high, but the length of the label as it goes around the entire "can" is given by the circumference of the circle base. The circumference is given by 2(pi)(2) = 12.56 units. Thus the rectangle area is 3(12.56) = 37.60 square units. Here's a puzzler for you, why are the area and the circumference the same number??? The yellow square pyramid can be decomposed into a square base and four congruent triangle faces. The area of the base is 2 X 2 = 4 square units. The area of one triangular face can be found after we identify the height of that face. Since the height generally given is an altitude (from the point at the top to the center of the square base), we will need to use the Pythagorean Theorem to find this height. Using the altitude as a leg, and half the length of the square base as the other leg, we find the hypotenuse (face height) as follows: 52 + 12 = 26. Taking the square root, we find the face height is 5.1 units. Then the triangular face has an area of 1/2(2)(5.1) = 5.1 square units. Since we have four triangular faces, we add four to the base area to get 4(5.1) + 4 = 24.4 square units. The pink pentagonal prism consists of two pentagons bases and five square sides. The square sides each have area 1 X 1 = 1 square unit, so five of them have an area of 5 square units. The pentagons can be divided into five congruent triangles inside each pentagon. The height of each triangle is .69 units, so their area is (1/2)(1)(.69) = .345 square units. For five triangles, the area would be 1.725 which we double for both bases to 3.45 and add to the five squares for a total surface area of 8.45 square units for the prism.

Challenge problem