VOLUME AND SURFACE AREA OF SOLIDS

VOLUME The amount of space a figure occupies.

Units for volume are cubic units: m3, ft3, cm3, etc.

SURFACE AREA The total area of all the surfaces of a figure. Surface area formulas come from adding up the

areas of each surface, using the usual area formulas to calculate (imagine unfolding a box).

Units for surface area are square units: m2, ft2, cm2, etc.

RECTANGULAR PRISM (BOX) Volume: V lwh

3

(3m)(2m)(5m)

30m [cubic meters]

V lwh

RECTANGULAR PRISM (BOX) Surface Area: 2 2 2SA lw wh lh

2 2 2 2

2 2 22(3m)(2m) 2(2m)(5m) 2(3m)(5m)

12m 20m 30m 62m [square meters]

SA lw wh lh

EXAMPLE What are the volume and surface area

of a rectangular prism with a length of 6 inches, width of 4 inches, and height of 10 inches?

3(6)(4)(10) 240 inV lwh

2

2 2 22(10)(4) 2(4)(6) 2(6)(10)

80 48 120 248 in

SA lw wh lh

CUBE The cube is just a special case of the box where

l=w=h. Since all the sides have equal length, you’ll sometimes see it given as x or s.

That means you’re really using the same formulas, but they’ve been modified to reflect that change.

3( )( )V lwh s s s s

2 2 2 2

2 2 22( )( ) 2( )( ) 2( )( )

2 2 62

SA lw wh lhs s s s s s

s ss s

EXAMPLE A cube has a volume of 64 cubic

inches. What is the length of each side, and what is the surface area of the cube?

3

3 64

64 4 in

s

s

2

2 2

6

6(4) 96 in

SA s

SA

CYLINDER Volume: Notice the formula comes

from the area of the circular base, times the height.

2V r h

CYLINDER Surface Area: The surface area formula comes from the

idea of peeling the label off a can – the circumference of the circle becomes the top edge of the label.

22 2rhSA r

CYLINDER That plus the top and bottom circle

areas gives the total surface area:

22 2rhSA r

EXAMPLE What are the volume and

surface area of the cylinder shown to the right? 2

2 3(7) 112(4) 352 in

V r h

2

2

2

2

(4)(7) 2 (

2

2

56

4)

32 88 276 in

rh rSA

PRISMS IN GENERAL The box and the cylinder are both shapes

that stand upright on their base, with the sides perpendicular to the floor.

The general name for this type of shape is a right prism, and we can make a general statement about the formula for volume…

PRISMS IN GENERAL Recall that for the box, V = lwh. Also

notice that the area of the base of the box is A = lw. So we can say the volume of a box is given by V=Ah, where A is the area of the base.

For the cylinder, we saw V=πr2h. But the area of the circular base is A = π r2. So the volume of a cylinder could be given as V=Ah, where A is the area of the base.

PRISMS IN GENERAL That turns out to be the general rule –

the volume of any right prism can be calculated from V=Ah, as long as we can calculate the area of the base.

EXAMPLE What would the

formula be for the volume of a prism with a triangular base, as shown to the right?1

2A bk

12

V Ah bkh

PYRAMID (SQUARE BASE) Volume: 21

3V s h

PYRAMID (SQUARE BASE) Surface area: Again, surface area

comes from picking apart the figure and finding the area of each surface.

e is called “slant height” and is the height of the triangle that forms the side of the pyramid

2 142

SA s se

2 214

she

PYRAMID (SQUARE BASE)

2A s 2 2 2

2 2

1 )(214

h

e h s

e s

2

41A se

2 142

SA s se

EXAMPLE What are the volume and

surface area of the pyramid shown to the right?

2 2 31 1 (4) (10)3 3

53.33 ftV s h

2 2 2 21 1 (4)4 4

10 104 10.20he s

2 2 214 (4) 972(4)(10.20)2

.6 ftSA s se

CONE Volume: 21

3V hr

CONE Surface area: As with the pyramid, e is the slant

height or edge height, and its formula comes from the Pythagorean Theorem

2S r reA

2 2e r h

2r

re

EXAMPLE What are the volume and

surface area of a cone with the dimensions shown in the figure?

2 2 3(10)1 1 (30) 3142 cm3 3

rV h

2 2 2 210 30 1000 31.6he r

2 2 2(10) (10)(31.6) 1307 cmrS eA r

PYRAMID (GENERAL BASE) We can make the same observation and

generalization about pyramids (and a cone is just a pyramid with a circular base) that we did about cylinders: if we know the area A of the base of the pyramid, the volume can be calculated from

13

V Ah

SPHERE Volume: Surface area:

343

V r

24SA r

EXAMPLE What are the volume and surface area

of a sphere with radius 2 ft?

3 3 3(24 ) 33.53

ft43

rV

2 2 2(2) 50.3 ft4 4rSA

THE END