Download Mod 3 - Aim #12 - Manhasset Public Schools

CC Geometry H
Aim #12: How do we compute the volume and surface area of a sphere?
Do Now:
1. Complete: A circle with center C and radius r is the set of all points in a plane a
given distance r, called the ______________, from a given point C, called the
_______________.
2. Picture a marble and a beach ball. Which one would you describe as a sphere?
___________What is a significant difference between the two figures?
Definition: Given a point C in three-dimensional space and a number r > 0, the sphere
with center C and radius r is the set of all points in space that are a distance r from
Examples of spheres: beach ball, soap bubble
the point C.
Definition: Given a point C in three-dimensional space and a number r > 0, the solid
sphere with center C and radius r is the set of all points in space whose distance
from the point C is less than or equal to r.
Examples of solid spheres include: marble, planet
A sphere is _________________ while a solid sphere is _________________
The term hemisphere refers to a half-sphere, and solid hemisphere refers to a
solid half-sphere.
Formula for Volume of a Sphere
Formula for Surface Area of a Sphere
SA = 4πr
where r = length of the radius
Examples:
1) Write a formula for the volume of the hemisphere.
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2) Find the surface area and volume of a sphere with a diameter of 12 cm. Give an
exact answer and rounded to the nearest tenth.
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3) Find, in terms of π, the volume of a sphere whose surface area is 324π cm .
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4) Find, in terms of π, the surface area of a sphere whose volume is 2304π cm .
5) Find, to the nearest tenth, the surface area of a sphere whose volume is
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7238.22 cm .
6) Find the volume of the composite solid, to the nearest tenth.
7) An ice cream cone is 12 cm deep and 5 cm across the opening of the cone. Two
hemisphere shaped scoops of ice cream, which also have diameters of 5 cm, are
placed on top of the cone. If the ice cream were to melt into the cone, will it
overflow? Justify your answer.
8) Bouncy rubber balls are hollow composed of a rubber shell 0.4 inches thick and
an outside diameter of 3.2 inches. The price of the rubber needed to produce this
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toy is $0.035/in .
a) What is the cost of producing one case, which holds 50 rubber balls? Round to
the nearest cent.
0.4
3.2
b) If each ball is sold for $0.75, how much profit is earned on each ball sold, to
the nearest penny?
9) Snow globes consist of a glass sphere that is filled with liquid and other
contents. If the inside radius of the snow globe is 3 inches, find the approximate
volume of material, to the nearest cubic inch, that can fit inside.
10) Consider a right circular cylinder with radius r and height h. The area of
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each base is πr . Think of the lateral surface area as a label on a soup can. If
you make a vertical cut along the label and unroll it, the label unrolls to the shape
of a ___________________.
r
h
a. Find the dimensions of the rectangle.
b. What is the lateral (or curved) area of the cylinder?
c. What is the total surface area of the cylinder?
d. Find the lateral and total surface area of a cylinder with diameter 6 and
height 8.4, in terms of π.
Let's Sum it Up!
Volume of a Sphere:
Surface Area of a Sphere:
Name_______________________
Date __________________
CC Geometry H
HW # 12
1) A solid sphere has volume 36π. What is the radius?
2) A sphere has surface area 16π. What is the radius?
3) Find, to the nearest tenth, the surface area of a sphere whose volume is
3
972π cm .
4) A sphere and a circular cylinder have the same radius, r, and the height of the
cylinder is 2r.
a. What is the ratio of the volumes of the solids?
b. What is the ratio of the surface area of the sphere to the total surface area
of the cylinder?
5) A semicircular disk of radius 3 feet is revolved about its diameter (straight
side) one complete revolution. Describe the solid determined by this revolution, and
then find the volume of the solid, in terms of π.
6) The radii of two spheres are 5 and 8. What is the ratio of the surface areas?
__________ the volumes? _________
Review
1) The radius of each circle is 1 cm. The hypotenuse of the
isosceles right triangle measure
cm. Find the area
inside the triangle, but outside the circles. 2) A right circular cone has radius 5 cm and slant height 13 cm. Find the volume,
in terms of π.
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3) The volumes of two similar cylinders are 512 cm and 2197 cm . What is the
ratio of their heights? __________ their lateral areas? __________
4) If a rectangular pyramid is sliced parallel to its base, the remaining portion of
the pyramid is the frustum. Find its volume.
12 m
20 m
9 m
15 m
6 m
m
10
5) Which figure has the greatest area? Justify your answer.
60
0
15 m
26
6) Find the volume of the cone in terms of π:
26 in.
20 in