Exercise 54 Page 838

The given solid is a pyramid.

To calculate the lateral area of a pyramid, we can use the known formula where P is the perimeter of the base and l is the slant height. L=1/2PlThe base of the pyramid is a regular hexagon with the side length of 10.5in. Let's calculate its perimeter!

P = 6s

Substitute

s= 10.5

P = 6( 10.5)

Multiply

P = 63

We are also given that the slant height l is 15in. Let's substitute all of these values into the formula for the lateral area and calculate it.

L = 1/2 P l

SubstituteII

P= 63, l= 15

L = 1/2 ( 63)( 15)

Multiply

L = 472.5

We have found that the lateral area is 472.5in^2. To find the surface area, all we have to do is add the base area B to the lateral area L. The base is a regular hexagon with a side length of 10.5in. Let's find the length of the apothem and the area of the base.

A central angle of a hexagon is 60^(∘), so the angle formed in the triangle above is 30^(∘). We can use the tangent function to find the apothem a.

tan30^(∘) = 5.25/a

▼

Solve for a

MultEqn

LHS * a=RHS* a

atan30^(∘) = 5.25

DivEqn

.LHS /tan30^(∘).=.RHS /tan30^(∘).

a = 5.25/tan30^(∘)

UseCalc

Use a calculator

a = 9.093266...

RoundDec

Round to 1 decimal place(s)

a = 9.1

Now that we know the apothem, we can calculate the area of the base using the following formula for the area of a regular polyhedron, where P is the perimeter and a is the apothem. B = 1/2 Pa Let's substitute 30 for P and 9.1 for a and calculate B.

B = 1/2 Pa

SubstituteII

P= 30, a= 9.1

B = 1/2( 30)( 9.1)

Multiply

B = 136.5

We can now calculate the surface area.

S = L + B

SubstituteII

L= 472.5, B= 136.5

S = 472.5+ 136.5

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