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Volume and Surface Area of Three-dimensional Figures

Three-dimensional Figures

The most common three-dimensional figures are: prisms, cylinders, pyramids, cones and spheres. They all have two important qualities: volume and surface area. Here is a list containing the figures and their properties.

Prism

Prism and formulas for calculation

Example 1

A prism has a triangle with the area B = 12cm2 as its base, and height h = 5cm. Calculate the volume of the prism.

You’ll find the volume by multiplying the area of the base B with the height h:

Volume = B h = 12cm2 5cm = 60cm3.

Cylinder

Formula of the surface area of cylinder

Example 2

A cylinder has the radius r = 2.0cm and height h = 8.0cm. Calculate the volume and the surface area of the cylinder.

Volume

The formula for the volume of a cylinder is V = B h. You have the height h, but have to find the base area B. Since the base area in a cylinder is a circle, you’ll have to find the area of a circle:

B = πr2 = π 2.0cm 2.0cm 12.6cm2.

Now you’ll find the volume of a cylinder by multiplying the base area B with the height h:

Volume = B h 12.6cm2 8.0cm 101cm3.

Volume = B h 12.6cm2 8.0cm 101cm3.

Surface Area

To find the surface area of a cylinder, you add together the areas of all the sides. The green rectangle you see on the drawing comes from cutting and folding out the walls of the cylinder. First, you calculate the area of the top and bottom of the cylinder (they have the same area), and then the area of the rectangle which is the height. The length of the rectangle is the circumference of the circle (see the figure): Area of a circle = πr2 = π 2.0cm 2.0cm 12.6cm2, Area of a rectangle = l h = 2πr h = 2 π 2.0cm 8.0cm 101cm2.

The surface area of a cylinder then becomes

Surface area = 2 Area of a circle + Area of a rectangle 2 12.6cm2 + 101cm2 = 126.2cm2.

Surface area = 2 Area of a circle + Area of a rectangle 2 12.6cm2 + 101cm2 = 126.2cm2.

Pyramid

Pyramid and formulas for calculation

Example 3

A pyramid has a triangle with area B = 9cm2 as its base. The height h of the pyramid is 4cm. Calculate the volume of the pyramid.

You’ll find the volume of the pyramid by multiplying the area of the base area B with the height h, and the dividing by 3:

Volume = B h 3 = 9cm2 4cm 3 = 12cm3.

Cone

Formula of the surface area of cone

Example 4

A cone has a circle with radius r = 3.0cm as its base. The height h of the cone is 7.0cm and side s = 9.0cm. Calculate the volume and the surface area of the cone.

Volume

The volume has the formula V = B h, so you first need to find the area of the base B. The area of the circle is:

B = πr2 = π 3.0cm 3.0cm 28.3cm2.

Then you’ll find the volume of the cone by multiplying the area of the base B with the height h, and then divide by 3: Volume = B h 3 = 28.3cm2 7.0cm 3 66.0cm3.

Surface Area

You’ll find the surface area by using the formula for the surface area of a cone. Here you can put your values directly in to the formula and calculate.

Surface area = πr2 + πrs = π 3.0cm 3.0cm + π 3.0cm 9.0cm 113cm2.

Surface area = πr2 + πrs = π 3.0cm 3.0cm + π 3.0cm 9.0cm 113cm2.

Sphere

Sphere and formulas for calculation

Example 5

A sphere has radius 5.0cm. Calculate the volume and the surface area of the sphere.

Volume

You1’ll find the volume by using the formula for the volume of a sphere: Volume = 4πr3 3 = 4 π 5.0cm 5.0cm 5.0cm 3 524cm3.

Surface Area

You’ll find the surface area by using the formula for the surface area of a sphere: Surface area = 4πr2 = 4 π 5.0cm 5.0cm 314cm2.

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