SURFACE AREA & VOLUME

Spheres

A sphere is a shape where the distance from the center to the edge is the same in all directions. This distance is called the radius ( r ).

Cylinders

A cylinder is a prism with a circular base of radius (r). The height (h) is the distance from the bottom base to the top of the cylinder.

Cones

A cone is a pyramid with a circular base of radius (r) and the side length (s) is the length of the side. The height (h) is the distance from the center of the base to the top of the cone.

Pyramids

A pyramid is a solid figure with a polygonal base and triangular faces that meet at a common point over the center of the base.The height (h) is the distance from the base to the apex or top of the pyramid.The side length (s) is the height of the face triangles.The area (A) of the base (b)is calculated according to the shape of the base.

Rectangular Prisms

A rectangular prism can be described as a stack of rectangles. The height (h) is the height of the prism, the width (w) is the width of the base, and the length (l) is the length of the base.

Triangular Prism

A prism can be described as a stack of shapes. The figure shows a prism of triangles stacked d thick, but any shape could be used. For a triangluar prism, the height (h) is the height of the base, the base (b) is the width of the base, and the length (l) is the length of the prism.

A = bh + 3bl

Application I

A standard 20-gallon aquarium tank is a rectangular prism that holds approximately 4600 cubic inches of water. The bottom glass needs to be 24 inches by 12 inches to fit on the stand.

Application I

Find the height of the aquarium to the nearest inch.

24 inches

12 inches

V = l x w x h 4600 = (24) x (12)

x h 4600 = 288 x h 16.0 inches = h

V = 4600 cubic inches

Application I

Find the total amount of glass needed in square inches for the six faces.

24 inches

12 inches

16 inches

TSA = 2(24)(12) + 2(12)(16) + 2(24)(16)

TSA = 576 + 384+768 TSA = 1728 in2

12 inches

24 inches

Application II

Does the can or the box have a greater surface area? a greater volume?

6 cm

9 cm9 cm

5 cm

5 cm

Cylinder Square Prism

Application II

6 cm

9 cm

Surface Area of can

rhrTSA 22 2 )9)(3(2)3(2 2 TSA

5418 TSA272 cmTSA 22.226 cmTSA

9 cm

6 cm

Application II

Surface Area of box

)4()2( 2 shsTSA ))9)(5(4()5*2( 2 TSA

18050TSA2230 cmTSA

9 cm

5 cm

5 cm

9 cm

5 cm

5 cm

Application II

Since 230cm2 > 288.2cm2, the box has a greater surface area.

Which container holds more juice?

Application II

6 cm

9 cm

Volume of can

hrV 2)9()3( 2V

)9)(9(V381 cmV 35.254 cmV

9 cm

6 cm

Application II

Volume of box

hwlV **9*5*5V

9*25V3225 cmV

9 cm

5 cm

5 cm

9 cm

5 cm

5 cm

Application II

Since 254.5cm3 > 225cm3, the can has greater volume.

If the can and the box were the same price, which one would you buy?

Application III

A birthday gift is 55 cm long, 40 cm wide, and 5 cm high. You have one sheet of wrapping paper that is 75 cm by 100 cm. Is the paper large enough to wrap the gift? Explain.

Application III

TSA = 2(55)(40)+2(55)(5)+2(40)(5)

TSA = 4400 + 550 + 400 TSA = 5350 sq cm

A = 75 x 100 A = 7500 sq

cm

5 cm

55 cm

40 cm

75 cm

100cm

Application IV

If both of these cans of pizza sauce are cylinders, which is the better buy?

20.5 cm

7.5 cm

11 cm

10 cm

$3.49

$1.09

Application IV

V = πr2h V = π(5)2(20.5) V = π * 25 * 20.5 V = 512.5 π cm3

V = 1610.1 cm3

= 461 cm3 per dollar

20.5 cm

10 cm

$3.49

49.3$

1.1610 3cm

Application IV

V = πr2h V = π(3.75)2(11) V = 154.7 π cm3

V = 486.0 cm3

= 446 cm3 per dollar09.1$

486 3cm

7.5 cm

11 cm

$1.09

Application IV

Which jar would you buy? Since 461 cm2 per dollar > 446 cm2

per dollar, you should buy the first jar.