Properties of Regular Hexagons

Oct 13, 2025 by miodeya.com 128 views

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Introduction

In this article, we will explore the relationship between the radii of circles inscribing two regular hexagons and use this relationship to find the perimeter, side length, and radius of the smaller hexagon. We will start by understanding the properties of regular hexagons and the relationship between their side lengths and radii.

Properties of Regular Hexagons

A regular hexagon is a polygon with six equal sides and six equal angles. Each interior angle of a regular hexagon measures 120 degrees. The side length of a regular hexagon is the distance between two adjacent vertices. The radius of a circle inscribing a regular hexagon is the distance from the center of the hexagon to one of its vertices.

Relationship Between Side Length and Radius

Let's consider a regular hexagon with side length s and radius r. We can draw a line from the center of the hexagon to one of its vertices, which is also the radius of the inscribed circle. We can also draw a line from the center of the hexagon to the midpoint of one of its sides, which is also the apothem of the hexagon. The apothem is the distance from the center of the hexagon to the midpoint of one of its sides.

Using the Pythagorean theorem, we can relate the side length s, the radius r, and the apothem a as follows:

s^2 = r^2 + a^2

Since the apothem is half the side length, we can substitute a = s/2 into the equation:

s^2 = r^2 + (s/2)^2

Simplifying the equation, we get:

s^2 = r^2 + s^2/4

Multiplying both sides by 4, we get:

4s^2 = 4r^2 + s^2

Subtracting s^2 from both sides, we get:

3s^2 = 4r^2

Dividing both sides by 3, we get:

s^2 = (4/3)r^2

Taking the square root of both sides, we get:

s = (2/√3)r

Relationship Between Radii of Inscribed Circles

We are given that the ratio of the radii of the circles inscribing the two regular hexagons is 7/5. Let's denote the radius of the larger hexagon as R and the radius of the smaller hexagon as r. We can write the ratio as:

R/r = 7/5

Finding the Perimeter of the Larger Hexagon

We are given that the perimeter of the larger hexagon is 63 cm. Since the perimeter of a regular hexagon is six times its side length, we can write:

6s = 63

Dividing both sides by 6, we get:

s = 63/6

s = 10.5

Finding the Radius of the Larger Hexagon

Using the relationship between the side length and radius, we can find the radius of the larger hexagon:

s = (2/√3)r

Rearranging the equation, we get:

r = (√3/2)s

Substituting s = 10.5, we get:

r = (√3/2)(10.5)

r = 5.75

Finding the Perimeter of the Smaller Hexagon

Using the ratio of the radii, we can find the radius of the smaller hexagon:

R/r = 7/5

Rearranging the equation, we get:

r = (5/7)R

Substituting R = 5.75, we get:

r = (5/7)(5.75)

r = 4.11

Now that we have the radius of the smaller hexagon, we can find its side length:

s = (2/√3)r

Substituting r = 4.11, we get:

s = (2/√3)(4.11)

s = 2.38

Finding the Perimeter of the Smaller Hexagon

Using the side length, we can find the perimeter of the smaller hexagon:

6s = 6(2.38)

6s = 14.28

Conclusion

In this article, we used the relationship between the side length and radius of a regular hexagon to find the perimeter, side length, and radius of the smaller hexagon. We started by understanding the properties of regular hexagons and the relationship between their side lengths and radii. We then used the ratio of the radii of the circles inscribing the two regular hexagons to find the radius of the smaller hexagon. Finally, we used the radius of the smaller hexagon to find its side length and perimeter.

Perimeter of the Smaller Hexagon

14.28 cm

Side Length of the Smaller Hexagon

2.38 cm

Radius of the Smaller Hexagon

4.11 cm

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Frequently Asked Questions

Q: What is the relationship between the side length and radius of a regular hexagon?

A: The relationship between the side length and radius of a regular hexagon is given by the equation:

s = (2/√3)r

where s is the side length and r is the radius.

Q: How do you find the radius of the larger hexagon?

A: To find the radius of the larger hexagon, you need to know its side length. Since the perimeter of the larger hexagon is given as 63 cm, you can find its side length by dividing the perimeter by 6:

s = 63/6

s = 10.5

Then, you can use the relationship between the side length and radius to find the radius:

r = (√3/2)s

Substituting s = 10.5, you get:

r = (√3/2)(10.5)

r = 5.75

Q: How do you find the radius of the smaller hexagon?

A: To find the radius of the smaller hexagon, you can use the ratio of the radii of the circles inscribing the two regular hexagons:

R/r = 7/5

Rearranging the equation, you get:

r = (5/7)R

Substituting R = 5.75, you get:

r = (5/7)(5.75)

r = 4.11

Q: How do you find the side length of the smaller hexagon?

A: To find the side length of the smaller hexagon, you can use the relationship between the side length and radius:

s = (2/√3)r

Substituting r = 4.11, you get:

s = (2/√3)(4.11)

s = 2.38

Q: How do you find the perimeter of the smaller hexagon?

A: To find the perimeter of the smaller hexagon, you can multiply its side length by 6:

6s = 6(2.38)

6s = 14.28

Q: What is the significance of the ratio of the radii of the circles inscribing the two regular hexagons?

A: The ratio of the radii of the circles inscribing the two regular hexagons is 7/5. This ratio is used to find the radius of the smaller hexagon.

Q: What is the relationship between the side length and radius of a regular hexagon?

A: The relationship between the side length and radius of a regular hexagon is given by the equation:

s = (2/√3)r

Q: How do you find the perimeter of a regular hexagon?

A: To find the perimeter of a regular hexagon, you can multiply its side length by 6:

6s = 6(2.38)

6s = 14.28

Q: What is the significance of the radius of a regular hexagon?

A: The radius of a regular hexagon is the distance from the center of the hexagon to one of its vertices.

Q: What is the significance of the side length of a regular hexagon?

A: The side length of a regular hexagon is the distance between two adjacent vertices.

Q: How do you find the apothem of a regular hexagon?

A: To find the apothem of a regular hexagon, you can use the relationship between the side length and radius:

a = s/2

Substituting s = 10.5, you get:

a = 10.5/2

a = 5.25

Q: What is the significance of the apothem of a regular hexagon?

A: The apothem of a regular hexagon is the distance from the center of the hexagon to the midpoint of one of its sides.

Q: How do you find the area of a regular hexagon?

A: To find the area of a regular hexagon, you can use the formula:

A = (3√3/2)s^2

Substituting s = 10.5, you get:

A = (3√3/2)(10.5)^2

A = 173.25

Q: What is the significance of the area of a regular hexagon?

A: The area of a regular hexagon is the amount of space inside the hexagon.

Q: How do you find the volume of a regular hexagonal prism?

A: To find the volume of a regular hexagonal prism, you can use the formula:

V = (3√3/2)s^2h

Substituting s = 10.5 and h = 5, you get:

V = (3√3/2)(10.5)^2(5)

V = 1732.5

Q: What is the significance of the volume of a regular hexagonal prism?

A: The volume of a regular hexagonal prism is the amount of space inside the prism.

Q: How do you find the surface area of a regular hexagonal prism?

A: To find the surface area of a regular hexagonal prism, you can use the formula:

SA = 6s^2 + 6sh

Substituting s = 10.5 and h = 5, you get:

SA = 6(10.5)^2 + 6(10.5)(5)

SA = 648

Q: What is the significance of the surface area of a regular hexagonal prism?

A: The surface area of a regular hexagonal prism is the amount of space on the surface of the prism.

Q: How do you find the lateral surface area of a regular hexagonal prism?

A: To find the lateral surface area of a regular hexagonal prism, you can use the formula:

LSA = 6sh

Substituting s = 10.5 and h = 5, you get:

LSA = 6(10.5)(5)

LSA = 315

Q: What is the significance of the lateral surface area of a regular hexagonal prism?

A: The lateral surface area of a regular hexagonal prism is the amount of space on the sides of the prism.

Q: How do you find the area of the base of a regular hexagonal prism?

A: To find the area of the base of a regular hexagonal prism, you can use the formula:

A = (3√3/2)s^2

Substituting s = 10.5, you get:

A = (3√3/2)(10.5)^2

A = 173.25

Q: What is the significance of the area of the base of a regular hexagonal prism?

A: The area of the base of a regular hexagonal prism is the amount of space inside the base.

Q: How do you find the height of a regular hexagonal prism?

A: To find the height of a regular hexagonal prism, you can use the formula:

h = (2/√3)s

Substituting s = 10.5, you get:

h = (2/√3)(10.5)

h = 6.05

Q: What is the significance of the height of a regular hexagonal prism?

A: The height of a regular hexagonal prism is the distance between the base and the top of the prism.

Q: How do you find the slant height of a regular hexagonal prism?

A: To find the slant height of a regular hexagonal prism, you can use the formula:

SLH = √(h^2 + (s/2)^2)

Substituting h = 6.05 and s = 10.5, you get:

SLH = √(6.05^2 + (10.5/2)^2)

SLH = 7.35

Q: What is the significance of the slant height of a regular hexagonal prism?

A: The slant height of a regular hexagonal prism is the distance from the center of the base to the top of the prism.

Q: How do you find the volume of a regular hexagonal pyramid?

A: To find the volume of a regular hexagonal pyramid, you can use the formula:

V = (3√3/2)s^2h

Substituting s = 10.5 and h = 5, you get:

V = (3√3/2)(10.5)^2(5)

V = 1732.5

Q: What is the significance of the volume of a regular hexagonal pyramid?

A: The volume of a regular hexagonal pyramid is the amount of space inside the pyramid.

Q: How do you find the surface area of a regular hexagonal pyramid?

A: To find the surface area of a regular hexagonal pyramid, you can use the formula:

SA = (3√3/2)s^2 + 3s√(h^2 + (s/2)^2)

Substituting s = 10.5 and h = 5, you get:

SA = (3√3/2)(10.5)^2 + 3(10.5)√(5^2 + (10.5/2)^2)

SA = 1732.5 + 315

SA = 2047.5

Q: What is the significance of the surface area of a regular hexagonal pyramid?

A: The surface area of a regular hexagonal pyramid is the amount of space on the surface of the pyramid.