bact to solid geometry video lessons

Find the area of a sphere

Question:

ABC is an equilateral triangle. each side of the equilateral triangle is 3. The vertices of the equilateral triangle lie on a sphere. If the distance from the center of the sphere to the triangle plane is 2, then what is the surface area of the sphere?

what is the surface area of a sphere if all vertices of an equilateral triangle lie on a sphere?

Solution:

The surface area of a sphere is: S = 4 pi R2, in which R is the radius of the sphere. Now we will find the radius of the sphere.

equilateral triangle example

ABC is an equilateral triangle. AB = BC = CA = 3. The height of the equilateral triangle is: AD = (square root of 3)/2) × AB = (3/2) (square root of 3)

O1 is the center of the equilateral triangle ABC. AO1 is the distance from vertex A to the center of the equilateral triangle. AO1 = (2/3) × AD = (2/3) × (3/2) × square root of 3 = square root of 3.

Because all the vertices of the equilateral triangle ABC lie on the sphere, the projection of the center of the sphere O to the plane ABC is O1. O1 is the center of the triangle ABC. Because OO1 is perpendicular to the plane ABC, and AO1 lies on the plane ABC, so, OO1 is perpendicular to AO1. Connect AO, AO is the radius of the sphere. triangle AOO1 is a right triangle.

In right triangle AOO1,
AO2 = OO12 + AO12
= 22 + (square root of 3)2
= 4 + 3 = 7
AO = R, so, R2 = 7

The surface area of the sphere is: S = 4 pi R2 = 4 pi × 7 = 28 pi.

Therefore, the surface area of the sphere is 28 pi. Watch the video for more details.