Download 12.5 Surface Area of Pyramids

12.5 Surface Area of Pyramids
By Tosha Williams, Taylor Clear, and
Jessica Strassle
Objectives
• Find lateral areas of regular pyramids
• Find surface areas of regular pyramids
Characteristics of Regular Pyramids
• All of the faces, except the base, intersect at
one point called the vertex.
• The base is always a polygon.
• The faces that intersect at the vertex are
called lateral faces and form triangles. The
edges of the lateral faces that have the vertex
as an endpoint are called lateral edges.
• The altitude is the segment from the vertex
perpendicular to the base.
Definitions to Know
• If the base of a pyramid is a regular polygon
and the segment with endpoints that are the
center of the base and the vertex is perp. to
the base, then the pyramid is called a regular
pyramid.
• The height of each lateral face is called the
slant height (l) of the pyramid.
Vertex
Lateral Edge
Lateral Face
Altitude
Slant Height
Base
Lateral Area of a Regular Pyramid
• Formula
L = ½ Pl
P is for Perimeter, L is for the lateral area, l is for
slant height.
Example 1:
• The roof of a birdhouse is a regular hexagonal
pyramid. The base of the pyramid has sides of
4 in., and the slant height of the roof is 12in. If
the roof is made of copper find the amount of
copper used for the roof.
L=1/2Pl
=1/2(24)(12)
=144
Lateral area of a
regular pyramid
P=24, l=12. (P=24
because The sides
of the base
measure 4, so the
perimeter is 6(4) )
Multiply.
Your Turn
Find the Lateral area of the regular pyramid.
Your Turn
• The answer is 58.2 ft squared.
Surface Area of Regular Pyramid
• Formula
T= ½ Pl + B
T is the surface area of a regular pyramid, B is
the area of the base.
Example 2:
Find the surface area of the square pyramid. To find the surface area, first find the
slant height of the pyramid. The slant height is the hypotenuse of a right triangle
with legs that are the altitude and a segment with a length that is one-half the side
measure of the base
.
c^2 = a^2 + b^2
Pythagorean Theorem
l^2 = 9^2 + 24^2
a=9, b=24, c=l
l= the square root of 657
Simplify.
Now find the surface area of a regular pyramid. The perimeter of the base
is 4(18) or 72 meters, and the area of the base is 18^2 or 324 square
meters.
T=1/2Pl + B
Surface area of a regular
pyramid
T=1/2(72) (Square root of 657) + 324
P=72, l = square root of 657,
B = 324
T
Use a calculator.
1246.8
Your Turn
Find the surface area of a regular pyramid.
Your Turn
• The answer is 340 cm squared
Example 3:
• Surface area of pentagonal Pyramid.
The altitude, slant height, and apothem form a right triangle. Use the Pythagorean
theorem to find the apothem. Let a represent the length of the apothem.
Use Pythagorean Theorem.
(17)^2 = a^2 + 15^2
b=15, c=17
8= a
Simplify.
Now find the length of the sides of the base. The central angle of the pentagon measures
360 /5 or 72 degrees. Let x represent the measure of the angle formed by a radius and
the apothem. Then, x = 72/2, or 36.
Use Trigonometry to find the length of the sides.
36
Tan 36degrees = .5s/8
8
8(tan 36degrees)= .5s
Multiply each side by 8
16(tan 36degrees)=s
Multiply each side by 2
11.6 = s
Use a calculator.
s
Next, find the perimeter and area of the base.
P= 5s
= 5(11.6) or 58.
B= ½ Pa
= ½ (58)(8) or 232
Finally, find the surface area.
T= ½ Pl + B
Surface area of a regular pyramid
= ½ (58)(17) + 232
P= 58, l= 17, B=232
= 726.5
Simplify.
The surface area is approximately 726.5 square inches.
Assignment:
• Page 663.
– 7-16, 18,19, 21-23, 27.
Five test questions
• 1) The Slant height of a regular Square
pyramid is 10ft and the sides of the base are
4ft, Find the lateral area.(Answer is 80 ft
squared)
• 2) The altitude of a regular Square pyramid is
14 and the side lengths are 25. Find the Slant
height and the surface area.(Slant height is
18.8 units squared and the Surface area is
1563.4 units squared.)
Five Test questions(Cont.)
• 3) Find the surface area of a regular pentagonal
pyramid if the sides are 10 and the slant height is
13. ( The answer is 497.0 units squared)
• 4) Find the surface area of a regular triangular
pyramid if the slant height is 3 and the sides are
5.(The answer is 33.3 units squared)
• 5)Find the surface area of a regular square
pyramid if the side is 8 and the slant height is
15.(The answer is 304 units squared)