8.3 Surface Area and Volume of Prisms and Pyramids

Surface Area

 

- Surface area is the number of square unites needed to cover the surface of a three-dimensional object

 

- To understand surface area, imagine you are wrapping any 3D object (ex. a box or sphere) in wrapping paper. The amount of wrapping paper needed to completely cover the object (without overlapping or leaving spaces out) is the surface area.

 

- Surface area can be determined by adding up the areas of the faces in the prism/pyramid.

Ex.

 

(Insert Pic)

 

For a traingular prism:

SA=(bh)+(lw)+(bw)+(hw)

b=3cm

h=2cm

w=7 cm

l=3.61 cm

 

For Hypotenuse:

a²+b²=c²

2²+3²=c²

4+9=c²

13=c²

c=3.61 cm

 

SA=(3x2)+(3.61x7)+(3x7)+(2x7)

SA=6+25.27+21+14

SA=66.27cm²



-We can also find the surface area of a square-based pyramid by adding the area of the base and the sum of the area of the 4 lateral faces

Ex.

 

(Insert Pic 2)

 

SA=2(bh)+ b²

SA=2(40x25)+(40)²

SA=2(1000)+1600

SA=2000+1600

SA=3600m²

 

Volume

 

- Volume is the amount of space an object occupies, measured in cubic units

-To find the volume of a prism, we use the formula V=l x w x h

 

- To find the volume of a triangular prism, we use the formula V=((bh)/2) x h or V=(base area) x (height)

Ex.

 

(Insert Pic 1)

 

V=((3x2)/2) x h

V=(6/2) x 7

V=3 x 7

V= 21 cm³

 

- To find the volume of a square-based pyramid, use the formula V=1/3(b²h)

Ex.

 

(Insert Pic 2)

 

To find the height:

 

- Right now, the height of the pyramid remains unknown. However, we do know that the slant height (length of the side of the pyramid) is 25 m, and that the edge of the base to the centre of the pyramid is 20 m (40m/2)

 

- With this information, we can use the Pythagorean Theorem to find the height

 

c²-b²=a²

(In this formula, the height is represented by variable 'a')

 

c²-b²=a²

25²-20²=a²

625-400=a²

225=a²

a=15 m

 

Thus, the height of the pyramid is 15 m. We can now find the volume of this pyramid.

 

V=1/3(b²h)

V=1/3(b²h)

V=1/3 (40² x 15)

V=1/3(1600 x 15)

V=1/3 (24 000)

V= 8 000m³