Exercise 49 Page 616

We are given that in the figure an equilateral triangle lies inside a square, which is inside a regular pentagon, which is inside a regular hexagon. Let's take a look at the given diagram. Notice that all figures have side lengths of 8 units.

To evaluate the approximate area of the entire shaded region we should find the area of each figure. Let A_h, A_p, A_s, and A_t represent the areas of the hexagon, pentagon, square, and triangle, respectively. A=A_h-A_p+A_s-A_t We will evaluate each area one at a time. Let's start with the area of a hexagon.

The Area of the Hexagon

Let's focus on the hexagon. Notice that the regular hexagon can be divided into six congruent equilateral triangles.

Let's draw an apothem of this hexagon a. Since the central angle is 60^(∘) and the apothem bisects this angle and the corresponding side, we can use trigonometric ratios to find a.

Let's write and solve an equation using the tangent ratio.

tan 30^(∘)=4/a

MultEqn

LHS * a=RHS* a

atan30^(∘)=4

DivEqn

.LHS /tan30^(∘).=.RHS /tan30^(∘).

a=4/tan30^(∘)

UseCalc

Use a calculator

a=6.9282

RoundDec

Round to 1 decimal place(s)

a≈ 6.9

The apothem of the hexagon is approximately 6.9 units. Now let's recall the formula for the area of a regular polygon. A=1/2ans In this formula a is the apothem, n is the number of sides, and s is the side length. Let's substitute 6.9 for a, 6 for n, and 8 for s.

A_h=1/2ans

SubstituteValues

Substitute values

A_h=1/2( 6.9)( 6)( 8)

Multiply

A_h=165.6

The area of the hexagon is approximately 165.6 square units.

The Area of the Pentagon

Now let's move to the pentagon. Notice that the regular pentagon can be divided into five congruent triangles.

Let's draw an apothem of this pentagon a. Since the central angle is 72^(∘) and the apothem bisects this angle and the corresponding side, we can use the trigonometric ratios to find a.

Let's write and solve an equation using the tangent ratio.

tan 36^(∘)=4/a

MultEqn

LHS * a=RHS* a

atan36^(∘)=4

DivEqn

.LHS /tan36^(∘).=.RHS /tan36^(∘).

a=4/tan36^(∘)

UseCalc

Use a calculator

a=5.5055

RoundDec

Round to 1 decimal place(s)

a≈ 5.5

The apothem of the pentagon is approximately 5.5 units. Now let's recall the formula for the area of a regular polygon. A=1/2ans In this formula a is the apothem, n is the number of sides, and s is the side length. Let's substitute 5.5 for a, 5 for n, and 8 for s.

A_p=1/2ans

SubstituteValues

Substitute values

A_p=1/2( 5.5)( 5)( 8)

Multiply

A_p=110

The area of the pentagon is approximately 110 square units.

The Area of the Square

Let's recall the formula for the area of a square. A=s^2 In this formula s is the side length. Let's substitute 8 for s in the above formula to find the area of the square.

A_s=s^2

Substitute

s= 8

A_s= 8^2

CalcPow

Calculate power

A_s=64

The area of the square is 64 square units.

The Area of the Triangle

Finally, we will evaluate the area of the equilateral triangle. Let's notice that the height in this triangle divides the figure into two congruent 30-60^(∘)-90^(∘) triangles.

Let's recall that in 30-60^(∘)-90^(∘) triangle the longer leg is sqrt(3) times the shorter leg, which is 4. Therefore the longer leg of this triangle — which is also the height of the equilateral triangle — is 4sqrt(3).

Next let's recall that the area of a triangle is half of the product of the height and the corresponding side length. Let's substitute appropriate values and find the area of our triangle.

A_t=1/2hs

SubstituteII

s= 8, h= 4sqrt(3)

A_t=1/2( 4sqrt(3))( 8)

Multiply

A_t=16sqrt(3)

UseCalc

Use a calculator

A_t=27.7128...

RoundDec

Round to 1 decimal place(s)

A_t≈ 27.7