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McGraw Hill Integrated II, 2012 View details

5. Volumes of Pyramids and Cones

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Exercise 50 Page 847

Find the area of a regular hexagon.

About 54.35 square inches

Practice makes perfect

Let's analyze the given shaded region. Divide the regular hexagon into six congruent equilateral triangles.

We are asked to find its area. Notice that it is equal to the difference between the area of the circle and the area of the regular hexagon. c Area of Region = c Area of Circle - c Area of Hexagon The radius of the circle is half of its diameter. Therefore, r= 202=10 inches. Now we can use the formula for the area of a circle.

A_\text{circle}=\pi r^2

Substitute

r= 10

A_\text{circle}=\pi ({\color{#009600}{10}})^2

CalcPowProd

Calculate power and product

A_\text{circle}=100\pi

The area of the circle is 100π square inches. We will find the area of the hexagon. The hexagon consists of 6 equilateral triangles. Each side of the triangle is s=10 cm. Let's use the formula the area of an equilateral triangle.

A_\text{hexagon}=6A_\Delta

▼

Substitute values and evaluate

Substitute

A_(Δ)= s^2sqrt(3)/4

A_\text{hexagon}=6\cdot{\color{#0000FF}{\dfrac{s^2\sqrt{3}}{4}}}

Substitute

s= 10

A_\text{hexagon}=6\cdot\dfrac{({\color{#009600}{10}})^2\sqrt{3}}{4}

CalcPow

Calculate power

A_\text{hexagon}=6\cdot\dfrac{100\sqrt{3}}{4}

CalcQuot

Calculate quotient

A_\text{hexagon}=6\cdot 25\sqrt{3}

Multiply

A_\text{hexagon}=150\sqrt{3}

Therefore, the area of the hexagon is 150sqrt(3) square inches. Finally, let's find the area of the shaded region. c Area of Region = c Area of Circle - c Area of Hexagon ⇓ c Area of Region = 100π- 150sqrt(3) This tells us that the area of the shaded region is 100π-150sqrt(3) square inches. Let's approximate the answer.

A_\text{region}=100\pi - 150\sqrt{3}

UseCalc

Use a calculator

A_\text{region}=54.351644\ldots

RoundDec

Round to 2 decimal place(s)

A_\text{region}\approx 54.35

The area of the shaded region is about 54.35 square inches.

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Exercises p.844-847

Exercise 50 Page 847

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