The sphere and the cylinder have equal heights.

a right triangle is created by the radius of the sphere r, the radius of the cross section y, and the distance between the center of the sphere and the center of the cross section x.

so y2=r2−x2

by the pythagorean theorem.

the area of the cross section of the sphere, and every cross section parallel to it, is .

each cross section of the cylinder with two cones removed is the shape of an annulus with an area of πr2−πx2

.

by cavalieri's principle, this sphere and this cylinder with two cones removed have volumes.

the volume of the cylinder with radius r and height 2r is 2πr3

.

the volume of each cone with radius r and height r is 13πr3

.

the volume of the cylinder with two cones removed is 43πr3

.

therefore, the volume of the sphere is .how is the formula for the volume of a sphere derived?

a sphere and a cylinder with two cones removed. the sphere is on the left. the radius of the sphere is r and the height of the sphere is 2 r. a cross section of the sphere is drawn. the radius of the cross section is drawn. a line segment connects the center of the sphere to the center of the cross section. a second radius is drawn to create a right triangle with the radius of the cross section, labeled y, and the line segment that connects the centers of the sphere and cross section, labeled x. the cylinder with two cones removed is on the right. each cone shares a base with the cylinder. the vertex of each cone is the center of the cylinder. the radius of the cylinder is r and the height of the cylinder is 2 r. a cross section of the solid is drawn. it is the area between two concentric circles. the radius of the outer circle is r and the radius of the inner circle is x.

drag and drop the correct word into each box to complete the explanation.

the sphere and the cylinder have equal heights.

a right triangle is created by the radius of the sphere r, the radius of the cross section y, and the distance between the center of the sphere and the center of the cross section x.

so y2=r2−x2

by the pythagorean theorem.

the area of the cross section of the sphere, and every cross section parallel to it, is .

each cross section of the cylinder with two cones removed is the shape of an annulus with an area of πr2−πx2

.

by cavalieri's principle, this sphere and this cylinder with two cones removed have volumes.

the volume of the cylinder with radius r and height 2r is 2πr3

.

the volume of each cone with radius r and height r is 13πr3

.

the volume of the cylinder with two cones removed is 43πr3

.

therefore, the volume of the sphere is .
how is the formula for the volume of a sphere derived?

a sphere and a cylinder with two cones removed. the sphere is on the left. the radius of the sphere is r and the height of the sphere is 2 r. a cross section of the sphere is drawn. the radius of the cross section is drawn. a line segment connects the center of the sphere to the center of the cross section. a second radius is drawn to create a right triangle with the radius of the cross section, labeled y, and the line segment that connects the centers of the sphere and cross section, labeled x. the cylinder with two cones removed is on the right. each cone shares a base with the cylinder. the vertex of each cone is the center of the cylinder. the radius of the cylinder is r and the height of the cylinder is 2 r. a cross section of the solid is drawn. it is the area between two concentric circles. the radius of the outer circle is r and the radius of the inner circle is x.

drag and drop the correct word into each box to complete the explanation.

the sphere and the cylinder have equal heights.

a right triangle is created by the radius of the sphere r, the radius of the cross section y, and the distance between the center of the sphere and the center of the cross section x.

so y2=r2−x2

by the pythagorean theorem.

the area of the cross section of the sphere, and every cross section parallel to it, is .

each cross section of the cylinder with two cones removed is the shape of an annulus with an area of πr2−πx2

.

by cavalieri's principle, this sphere and this cylinder with two cones removed have volumes.

the volume of the cylinder with radius r and height 2r is 2πr3

.

the volume of each cone with radius r and height r is 13πr3

.

the volume of the cylinder with two cones removed is 43πr3

.

therefore, the volume of the sphere is .