Solids of revolution

Learning Objectives Identify the three-dimensional solid (cylinder, cone, sphere) generated by revolving a two-dimensional shape (rectangle, right triangle, semicircle). Determine the dimensions (radius, height) of a solid of revolution based on the original two-dimensional shape and the axis of revolution. Visualize and sketch the resulting solid of revolution. Apply the correct formulas to calculate the volume of cylinders, cones, and spheres. Apply the correct formulas to calculate the surface area of cylinders, cones, and spheres. Solve problems by working backwards from a given volume or surface area to find a missing dimension. Ever wonder how a potter on a wheel can turn a lump of clay into a perfectly symmetrical vase? 🏺 You're about to learn the mathematical s...

TermDefinitionExample Solid of RevolutionA three-dimensional figure formed by rotating a two-dimensional shape around a fixed line.Rotating a rectangle around one of its sides creates a cylinder. Axis of RevolutionThe fixed line that a two-dimensional shape is rotated around to create a solid of revolution.If you spin a right triangle around one of its legs, that leg is the axis of revolution. Generating RegionThe two-dimensional shape that is rotated to form the solid of revolution.A semicircle is the generating region for a sphere. Cylinder (as a solid of revolution)A solid formed by revolving a rectangle about one of its sides.A 5x3 rectangle revolved around the side of length 5 creates a cylinder with a height of 5 and a radius of 3. Cone (as a solid of revolution)A solid formed by re...

Volume of a Cylinder V = \pi r^2 h Use this formula to find the volume of a cylinder, where 'r' is the radius of the circular base and 'h' is the height of the cylinder. Volume of a Cone V = \frac{1}{3} \pi r^2 h Use this formula to find the volume of a cone. Notice it is exactly one-third the volume of a cylinder with the same radius 'r' and height 'h'. Volume of a Sphere V = \frac{4}{3} \pi r^3 Use this formula to find the volume of a sphere, where 'r' is the radius of the sphere. Surface Area of a Cylinder SA = 2\pi rh + 2\pi r^2 Use this formula to find the total surface area of a cylinder. The first part (2πrh) is the area of the curved side, and the second part (2πr²) is the area of the two circular base...