1. 12.7 Surface Area of Spheres
Angela Isac
Abby Kern
1st hour

2. Objectives Recognize and define basic properties of spheres.
Find surface areas of spheres.

3. What is a Sphere?
A sphere is the locus of all points that are a given distance from a given point called the center.
To help visualize a sphere, consider infinitely congruent circles in space that all have the same point for their center.

4. Properties of Spheres
A radius of a sphere is a segment with endpoints that are the center of the sphere and a point on the sphere. In the figure, DA, DB, and DC are radii.
A chord of a sphere is a segment with endpoints that are points on the sphere. In the figure, GF and AB are chords.
A diameter is a chord that contains the center of the sphere. In the figure, AB is the diameter.
A tangent to a sphere is a line that intersects the sphere in exactly one point. In the figure, JH is a tangent to the sphere at point E.

5. Great Circles
When a plane intersects a sphere so that it contains the center of the sphere, the intersection is called a great circle.
A great circle has the same center as the sphere, and its radii are also radii of the sphere.

6. Hemispheres
Each great circle separates a sphere into two congruent halves, each called a hemisphere.
A hemisphere has a circular base.

7. Example #1
In the figure, O is the center of the sphere, and plane R intersects the sphere in circle A. If AO = 3cm and OB=10cm, find AB.

8. Example #1 Cont. Use the Pythagorean Theorem to solve for AB.
OB2 = AB2 + AO2 Pythagorean Theorem
102 = AB OB = 10, AO = 3
100 = AB Simplify.
91 = AB2 Subtract 9 from each side.
9.5 = AB Use a calculator.
Answer: AB is approximately 9.5cm.

9. Surface Area of Spheres
If a sphere has a surface area of T square units and a radius of r units, then T= 4 r2 This is simply saying that the surface area (T) of the sphere is 4 times the area of the great circle ( r2).

10. Example #2
Find the surface area of the sphere given the area of the great circle.

11. Example #2 Cont. Use the formula for surface area to solve.
T = 4 r2 Surface are of the sphere.
T = 4(201.1) r2 = 201.1
T = Multiply.
Answer: in2

12. Area of a Hemisphere
Since a hemisphere is half a sphere, to find its surface area, find half of the surface area of the sphere and add the area of the great circle. T = ½(4 r2) + r2

13. Example #3
Find the area of the hemisphere.

14. Example #3 Cont. Use the formula for the surface area of a
hemisphere to solve.
T = ½(4 r2) + r Surface are of a hemisphere
T = ½[4 (4.2)2] + (4.2) Substitution
T = Use a calculator.
Answer: 166.3cm2

15. Assignment
Pre-AP Geometry: Page 674 #