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Wize High School Grade 9 Math Textbook > 3D Geometry - Measurements

Surface Area & Volume of a Sphere

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Surface Area of a Sphere

A sphere is a solid 3D object where every point on its surface is the same distance from the center.

Surface Area

The surface area of a sphere is 4 times the area of the circular cross-section through the middle of the sphere (passing through the diameter)

S.A.sphere=4πr2\large\boxed{\text{S.A.}_\text{sphere}=4\pi r^2}

Practice: Surface Area of a Sphere

Find the surface area of this sphere.


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Volume of a Sphere

The volume of a sphere is 23\dfrac{2}{3} the volume of the cylinder with the same circular base and the same height.


V=23π (rcylinder)2×hcylinder\begin{array}{rcl} V&=&\dfrac{2}{3}\pi~ (r_\text{cylinder})^2\times h_\text{cylinder}\\ \end{array}

From the diagram, we see that the radius of the cylinder is the radius of the sphere, and the height of the cylinder is 2 times the radius of the sphere:

V=23π r2×(2r)\begin{array}{rcl} V&=&\dfrac{2}{3}\pi~r^2\times (2r)\\ \end{array}

Simplifying, we get the formula for the volume of a sphere with radius rr:

Vsphere=43πr3\large\boxed{V_\text{sphere}=\dfrac{4}{3}\pi r^3}

Practice: Volume of a Sphere

Find the volume of this sphere.



Practice: Sphere

The outside of a soccer ball is made up of leather that costs $0.01/cm2cm^2. If the volume of a spherical soccer ball is 5575 cm35575~cm^3, what is the cost of leather needed to make this soccer ball?
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Practice: Spheres

If the surface area and volume of a sphere are the same, what do we know about the radius of the sphere?