Volume and Surface Area of Prisms and Pyramids

Volume and Surface Area of Prisms and Pyramids

A pyramid is a polyhedron whose base is a polygon and whose faces are triangles that meet a common vertex.

Lateral faces are the faces of a prism or pyramid that are not bases. The surface area of a prism or pyramid is the sum of the areas of the faces.

Surface Area and Volume of Prisms

Rectangular Prism

V=l ×w ×h

Volume of a rectangular prism

S.A=lw+lh+wh+lw+lh+wh

S.A=2lw+2lh+2wh

S.A=2(lw+lh+wh)

Surface area of a rectangular prism.

Example 1: A large Toblerone bar has the dimensions shown. Determine the surface area and the volume of this prism. Triangular prism

Solution:

S.A=2Atriangle+3Arectangle

=2b ×h2+3l ×w

=bh+3×305 times

=6h+549

Find the height,h of the triangle using P.Th.

32+h2=62

9+h2=36

h2=36-9

h2=27

h=±5.2

So, h=5.2 cm

-S.A.=6h+549

=6(5.2)+549

=31.2+549

=580.2

The surface area of the triangular prism is 580.2 cm2

In general, V=base area × height

VT-P=base areaheight

=Atriangle×height

=b ×h2×height

=6 ×5.22×30.5

=3 ×5.2 ×30.5

=475.8

The volume of the triangular prism is 475.8 cm2

Surface Area and Volume of Pyramids

The volume of any pyramid can be determined using the formula,

V=13base areaheight

In a pyramid, the height of a lateral face is called the slant height.

Example 1: Determine the surface area of the following pyramid

S.A=Asquare+4Atriangle

=35 ×35+435 ×2782

=1225+2 ×35 ×27.8

=1225+1946

=3171

The surface of the square-based pyramid is 3171 m2.

In order to find the volume of the pyramid, we need to find the height of the pyramid using P.Th.

h2+17.52=27.82

h2=27.82-17.52

h2=772.84-306.25

h2=466.54

h=466.54

h=21.6 m

V=base area×height

V=13Asquare×height

=35 3521.63

=264603

=8820

The volume is 8820 m3.

Example 2: Determine the volume of the following squared based pyramid.

Regular Square pyramid

-Refer to the previous example.

Example 3: A cereal box has a volume of 3000 cm^3. If its length is 20 cm and its width is 5 cm, what is its height?

Solution:

V=l ×w ×h ←Sub. V1l and w

3000=20 ×5 ×h Solve for h

3000=100h

3000100=100h100

h=30

The height of the box is 30 cm.