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1 University of Waterloo Centre for Education in Faculty of Mathematics Mathematics and Computing Junior Math Circles March 10, 2010 3D Geometry II Opening Problem Three tennis ball are packed in a cylinder. The balls at the top and bottom are touching the top and bot- tom of the cylinder, respectively, and all three balls are touching the sides of the cylinder. The tennis balls have a radius of 3 cm. What is the volume of air in the tube? Surface Area of 3D Figures Definition: The surface area SA of a 3D figure is the sum of the areas of all of its faces and curved surfaces. The surface area of any figure is measured in square units (or units squared). Surface Area of a Prism The surface area of a prism is equal to the area of both bases plus the area of each face that joins the two bases together. Each of the faces that joins the two bases together will have a width equal to one of the sides of the base and length equal to the height of the prism. SA = (2 × Area of Base) + (h × Perimeter of Base) + Note that for a cylinder, the perimeter of the base is actually the circumference of the circle, 2πr. Example A chocolate bar is packaged in the cardboard equilateral triangular prism 2 shown below. What area of cardboard is required to create the box? 3.5 cm 20 cm 4 cm We know that this shape is a prism with an equilateral triangle base. 1 The area of the triangle is ×3.5 cm×4 cm=7 cm2 . 2 Since the triangle is equilateral, all of its sides measure 4 cm, so its perimeter is 3 × 4 cm= 12 cm. The surface area of the prism is therefore (2 × 7 cm2 ) + (20 cm × 12 cm) = 14 cm2 + 240 cm2 = 254 cm2 . So, 254 cm2 of cardboard is needed to create the box. Exercise 1 Find the surface area of the following prisms. 1) 2) 5 cm 12 cm 24 cm 2m 6m SA = 2 × π × 22 + 6 × 2 × π × 2 SA = 2(5 × 12 + 24 × 12 + 5 × 24) = 8π + 24π = 2(60 + 288 + 120) = 32π = 936 cm2 ≈ 100.5 m2 Surface Area of a Pyramid Definition: The slant height, s, of a triangular face of a pyramid is the height of the triangle, running from the base to the common vertex. This is different from the height as shown below. 3 height slant height The surface area of a pyramid is equal to the area of the base plus the area of each triangular face that meets at the common vertex. The area of each triangular face is equal to its slant height times the length of the side of the base. SA = Area of Base + Area of Triangular Faces For a cone, the slant height is the length of a line from the common vertex to any point on the edge of the circular base. For a cone, SA = πrs + πr2 Example An ice cream cone with a radius of 4 cm and a slant height of 15 cm is going to be dipped in chocolate. What is the surface area that the chocolate will cover? 4 cm 15 cm An ice cream cone does not have a flat base like other cones, and so we do not need to include πr2 in the surface area, since this is the area of the circular base. Therefore, the surface area of the cone will be πrs. SA = π × 4 cm×15 cm= π × 60 cm2 ≈ 188.5 cm2 . Exercise 2 The pyramid below has 4 identical triangular faces and a square base. Find its surface area. 4 5m 6m SA = 62 + 4( 12 × 5 × 6) = 36 + 60 = 96 m2 Surface Area of a Sphere To find the surface area of a sphere, try this activity at home. 1. Find an orange that is approximately spherical. Measure its circumference at the widest point and use this to find its radius. On a sheet of paper, draw circles with the same radius. 2. The area of each of these circles is πr2 . How many of these areas do you think is equal to the surface area of the orange? 2? 3? 4? 5? More? 3. Peel the orange, and lay the pieces of peel inside one of the circles you drew. Once you fill a circle, move on to another one. How many circles did the orange peel cover? The orange peel covers 4 circles. This means that the surface area of a sphere is equal to four times the area of a circle with the same radius. The surface area of a sphere is: SA = 4πr2 Example A ping pong ball has a radius of 2 cm. What

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