12-3 Study Guide and Intervention

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12-3 Study Guide and Intervention

Surface Areas of Pyramids and Cones

Lateral and Surface Areas of Pyramids A pyramid is asolid with a polygon base. The lateral faces intersect in a commonpoint known as the vertex. The altitude is the segment from thevertex that is perpendicular to the base. For a regular pyramid,the base is a regular polygon and the altitude has an endpoint atthe center of the base. All the lateral edges are congruent and allthe lateral faces are congruent isosceles triangles. The height of eachlateral face is called the slant height.

LateralAreaof
aRegularPyramid / ThelateralareaLofaregularpyramidisL=Pℓ,whereℓ
istheslantheightandPistheperimeterofthebase.
SurfaceAreaof
aRegularPyramid / ThesurfaceareaSofaregularpyramidisS=Pℓ+B,whereℓistheslantheight,Pistheperimeterofthebase,
andBistheareaofthebase.

Example: For the regular square pyramid above, find the lateral area andsurface area if the length of a side of the base is 12 centimeters and the height is8 centimeters. Round to the nearest tenth if necessary.

Find the slant height.

= 62 + 82 Pythagorean Theorem

= 100 Simplify.

ℓ= 10 Take the positive square root of each side.

Chapter 1218Glencoe Geometry

NAME ______DATE______PERIOD ______

L = Pℓ Lateral area of a regular pyramid

= (48)(10) P = 4 ⋅ 12 or 48, ℓ = 10

= 240 Simplify.

S = Pℓ + B Surface area of a regular pyramid

= 240 + 144Pℓ = 240, B = 12 · 12 or 144

= 384

Chapter 1218Glencoe Geometry

NAME ______DATE______PERIOD ______

The lateral area is 240 square centimeters, and the surface area is 384 square centimeters.

Exercises

Find the lateral area and surface area of each regular pyramid. Round to thenearest tenth if necessary.

1.2.

3.4.

12-3 Study Guide and Intervention(continued)

Surface Areas of Pyramids and Cones

Lateral and Surface Areas of Cones A cone hasa circular base and a vertex. The axis of the cone is thesegment with endpoints at the vertex and the center ofthe base. If the axis is also the altitude, then the cone is aright cone. If the axis is not the altitude, then the coneis an oblique cone.

LateralArea
of aCone / ThelateralareaLofarightcircularconeisL=πrℓ,whereris theradiusandℓistheslantheight.
SurfaceArea
of aCone / ThesurfaceareaSofarightconeisS=πrℓ+πr2,whereris theradiusandℓistheslantheight.

Example: For the right cone above, find the lateral area and surface area ifthe radius is 6 centimeters and the height is 8 centimeters. Round to the nearesttenth if necessary.

Find the slant height.

= + Pythagorean Theorem

= 100 Simplify.

ℓ= 10 Take the positive square root of each side.

Chapter 1219Glencoe Geometry

NAME ______DATE______PERIOD ______

L = πrℓ Lateral area of a right cone

= π(6)(10) r = 6, ℓ = 10

≈ 188.5 Simplify.

S = πrℓ + Surface area of a right cone

≈ 188.5 + π(62) πrℓ≈ 188.5, r = 6

≈ 301.6 Simplify.

Chapter 1219Glencoe Geometry

NAME ______DATE______PERIOD ______

The lateral area is about 188.5 square centimeters and the surface area is about301.6 square centimeters.

Exercises

Find the lateral area and surface area of each cone. Round to the nearest tenth ifnecessary.

1.2.

3.4.

Chapter 1219Glencoe Geometry