CONNECT: Volume, Surface Area 2. SURFACE AREAS OF SOLIDS If you need to know more about plane shapes, areas, perimeters, solids or volumes of solids, please refer to CONNECT: Areas, Perimeters – 1. AREAS OF PLANE SHAPES; CONNECT: Areas, Perimeters – 2. PERIMETERS OF PLANE SHAPES and CONNECT: Volume, Surface Area – 1. VOLUMES OF SOLIDS. You may also need to review Pythagoras’ Theorem – if so, please refer to CONNECT: Pythagoras’ Theorem The surface area of a 3D shape is “the total area of the outside” of the shape (De Klerk, 2007, p. 129). So, to work out the surface area of a prism, you need to work out the area of each face and add these areas together. Before we calculate the surface area of an example, have a think about the units we will use to measure surface area. Example: Calculate the surface area of this rectangular prism: If we draw in the edges that are blocked from sight by the other faces, we can see the shapes of all the faces: The faces are all rectangles and so each of their areas is A = l x w units2. For each of the top and bottom faces, the area is 3cm x 4cm = 12cm2. For each of the left and right faces, the area is 4cm x 12cm = 48cm2. For each of the front and back faces, the area is 3cm x 12cm = 36cm2. So the total surface area is 12 x 2 + 48 x 2 + 36 x 2 cm2 that is, 192cm2. The units are square units because we are measuring area. 1 Surface area of a cylinder Take an ordinary A4 sheet of paper. Now roll it into a cylinder so that the edges just meet. Do you realise that you have used a rectangle to make the curved surface of a cylinder? Place your cylinder so that it stands on its circular base. The length of the rectangle is now the circumference of the circle (base) and the width of the rectangle is now the height of the cylinder. (Or vice versa, depending on which way you rolled the cylinder, but that will not make any difference to our formula.) h h l l So the area of the curved surface of the cylinder is = 2 × units2. (2 is the length of the circumference, which is the same as l, the length of the rectangle.퐴 휋푟) ℎ 휋푟 If the cylinder is open at both ends, the total surface area of the cylinder is just the area of the curved surface. If it is closed at one end, we need to add the area of that circle and if it is closed at both ends, we need to add the area of both circles. So, the total surface area of a cylinder could be any of the following: = 2 × units2 ← open at both ends 2 OR 퐴 = 2휋푟 × ℎ + units ← open at one end 2 2 OR 퐴 = 2휋푟 × ℎ + 휋2푟× units ← closed at both ends. 2 퐴 휋푟 ℎ 휋푟 Example: Find the surface area of the closed cylinder over the page. 2 r=6cm h=10.3cm 2 We need to use the formula = 2 × + 2 × units because the cylinder is closed. 2 퐴 휋푟 ℎ 휋푟 2 2 So, A = 2 x x 6 x 10.3 + 2 x x 6 cm 2 Using a calculator휋 we obtain휋 614.495 523 cm , that is, approximately 614.496cm2. Surface Area of a Cone The curved surface area of a cone is a little more involved to calculate. We need to know more than just our usual measurements – the radius of the circular base and the perpendicular height of the cone. We also need to know the slant height (s) of the cone. Diagram retrieved January 31, 2013, from http://www.mathsteacher.com.au/year10/ch14_measurement/18_cone/20cone.htm 3 If we know the slant height of the cone, we can use the formula = units 2 to find 퐴the area휋푟푠 of the curved surface. So for an open cone, this formula gives the total surface area of the cone. But if we have a closed cone, we need to add the area of the circle as well. So, the total surface area of a cone will be: 2 = units ← open cone 2 퐴 = 휋푟푠 + units ← closed cone 2 Example:퐴 Find휋푟푠 the 휋surface푟 area of the following closed cone: s = 8.1cm r = 5.1cm Curved surface area = cm2 = 휋푟푠 x 5.1 x 8.1 cm2 2 = 휋129.779 192 5… cm 2 Area of base = cm 2 = 휋푟 x 5.12 cm2 2 = 휋81.712 824 92… cm 2 So the total area = 129.779 192 5… + 81.712 824 92… cm 2 = 211.492 017 4… cm Round this to approximately 211.5cm2 4 But what if we have a cone and do not know s? For example, find the total surface area of this closed cone: s h = 4.1 mm r = 3.2mm We know the radius and the perpendicular height of the cone, but we need to know the slant height. You can see that the radius, vertical height and slant height form the sides of a right-angled triangle, where the slant height (s) is the hypotenuse. By Pythagoras’ Theorem, = + 2 2 2 푠 =푟3.2 +ℎ 4.1 2 2 2 푠 = 27.05 To find s, take the square root of 27.05, so 27.05 = 5.200 961 45… So, s is 5.200 961 45… mm. (To make √the final answer as accurate as possible, don’t round yet.) 2 Now, the curved surface area = mm 2 =휋푟푠 x 3.2 x 5.200 961 45… mm 2 = 휋52.285 767 3…mm 2 and the area of the base = mm 2 2 = 휋푟 × 3.2 mm 2 2 = 32.169휋 908 77…mm 2 So the total surface area is 52.285 767 3… + 32.169 908 77…mm 2 = 84.455 676 07…mm Only now do we round and so the area is approximately 84.5mm2 5 If you need help with any of the Maths covered in this resource (or any other Maths topics), you can make an appointment with Learning Development through Reception: phone (02) 4221 3977, or Level 3 (top floor), Building 11, or through your campus. REFERENCES De Klerk, J. (2007). Illustrated Maths Dictionary. 4th ed. Pearson. Melbourne. 6 .