12.6 Surface Area and Volume of Spheres Unit V Day 6

Do Now List the formulas: Euler’s theorem Volume of prism Volume of cylinder Surface area of prism Surface area of cylinder Volume of pyramid Volume of cone

Finding the Surface Area of a Sphere We know that a circle is the set of all points in a plane that are a given distance from a point. A sphere is the set of all points in space that are a given distance (radius) from a given point (center).

Ex. 1: Finding the Surface Area of a Sphere (baseball) A baseball and its leather covering are shown. The baseball has a radius of about 1.45 inches. Estimate the amount of leather used to cover the baseball. a. S = 4π(1.45)2 ≈ 26.4º

Theorem 12.11: Surface Area of a Sphere The surface area of a sphere with radius r is S = 4r2.

Ex. 1: Finding the Surface Area of a Sphere Find the surface area of each sphere. When the radius doubles, does the surface area double? S = 4π*22 = 16π in.2 S = 4π*42 = 64π in.2 The surface area of the sphere in part (b) is four times greater than the surface area of the sphere in part (a) because 16π • 4 = 64π. So, when the radius of a sphere doubles, the surface area does not double. Note: It appears to quadruple, but this is not a proof of that fact. To prove it, we would need to use radii of r and 2r.

Finding the Volume of a Sphere Let n be the number of pyramids needed to fill the sphere. The volume of each pyramid is __________. The sum of the base areas is _____. This is the __________________, which is ______. You can approximate the volume V of the sphere as _____________. Imagine that the interior of a sphere with radius r is approximated by n pyramids as shown, each with a base area of B and a height of r, as shown. 1/3 Br nB; surface area; 4πr2 Volume of the sphere ≈ n*volume of pyramid = n*1/3 Br Since nB ≈ 4πr2, we can substitute to get V ≈ 1/3(4πr2)r = 4/3 r3

Theorem 12.12: Volume of a Sphere The volume of a sphere with radius r is S = 4/3 r3. 3

Ex. 4: Finding the Volume of a Sphere (ball bearings) To make a steel ball bearing, a cylindrical slug is heated and pressed into a spherical shape with the same volume. Find the radius of the ball bearing to the right. To find the volume of the slug, use the formula for the volume of a cylinder. V = r2h = (12)(2) = 2 cm3 To find the radius of the ball bearing, use the formula for the volume of a sphere and solve for r. V =4/3r3 2 = 4/3r3 6 = 4r3 1.5 = r3 1.14  r

Closure How can you find the surface area of a sphere? the volume of a sphere? Use the formulas S= 4πr2 and V = 4/3πr3