Regular Polygons in Models: Structure & Design

1) A circular garden is modeled using a regular 20-gon. Why is this an effective model?

The perimeter of the 20-gon gets very close to the circle’s circumference:

The 20-gon has diagonals:

The 20-gon can be split into trapezoids:

The 20-gon has equal sides but unequal angles:

The effectiveness of modeling a circular garden with a regular 20-gon lies in the fact that as the number of sides increases, the perimeter of the polygon approaches the circumference of the circle. Thus, a 20-gon serves as a good approximation of a circle.

Explanation

The effectiveness of modeling a circular garden with a regular 20-gon lies in the fact that as the number of sides increases, the perimeter of the polygon approaches the circumference of the circle. Thus, a 20-gon serves as a good approximation of a circle.

2) A company wants a logo shaped like a regular heptagon. Which property guarantees visual symmetry?

Equal diagonals

Congruent central angles

Congruent diagonals

Unequal angles

A regular heptagon has congruent central angles, which ensures that all sides and angles are equal, creating visual symmetry. This property is essential for maintaining the uniform appearance of the logo.

Explanation

A regular heptagon has congruent central angles, which ensures that all sides and angles are equal, creating visual symmetry. This property is essential for maintaining the uniform appearance of the logo.

3) A company designs stop signs shaped as regular octagons. Why is this shape effective?

It is equilateral only:

It is equiangular only:

It is both equilateral and equiangular:

It resembles a circle:

The octagonal shape is effective for stop signs because it is both equilateral and equiangular, which ensures that all sides and angles are equal. This uniformity makes the sign easily recognizable from various angles and distances, enhancing safety on the roads.

Explanation

The octagonal shape is effective for stop signs because it is both equilateral and equiangular, which ensures that all sides and angles are equal. This uniformity makes the sign easily recognizable from various angles and distances, enhancing safety on the roads.

4) A honeycomb cell is modeled as a regular hexagon. This design is efficient because:

Hexagons minimize perimeter for given area:

Hexagons maximize diagonals:

Hexagons are congruent to circles:

Hexagons cannot tessellate:

Hexagons are chosen for honeycomb structures because they provide the maximum area with the least amount of perimeter compared to other shapes. This efficiency reduces the amount of wax needed to build the honeycomb.

Explanation

Hexagons are chosen for honeycomb structures because they provide the maximum area with the least amount of perimeter compared to other shapes. This efficiency reduces the amount of wax needed to build the honeycomb.

5) A tile company produces regular hexagonal tiles for flooring. Which property ensures they tessellate without gaps?

Each interior angle is 120°

Each central angle is 90°

Each side equals the radius

Each diagonal is perpendicular

Regular hexagonal tiles tessellate without gaps because each interior angle measures 120°, allowing them to fit together perfectly at the edges. This property enables them to cover a surface completely, which is essential for flooring applications.

Explanation

Regular hexagonal tiles tessellate without gaps because each interior angle measures 120°, allowing them to fit together perfectly at the edges. This property enables them to cover a surface completely, which is essential for flooring applications.

6) A triangular traffic sign is modeled as an equilateral triangle with side length 60 cm. Which geometric property makes it recognizable as a regular polygon?

It has right angles:

All three angles are congruent at 60°:

Its diagonals are unequal:

It has one obtuse angle:

A regular polygon is defined as a polygon with all sides and angles equal. In an equilateral triangle, all three angles are congruent at 60°, which is a defining characteristic of regular polygons.

Explanation

A regular polygon is defined as a polygon with all sides and angles equal. In an equilateral triangle, all three angles are congruent at 60°, which is a defining characteristic of regular polygons.

7) A gazebo roof is modeled as a regular octagon with side length 5 m and apothem 6 m. Which calculation gives the amount of roofing material required?

5 × 8

1/2 × (5 × 8) × 6

(5 × 8)²

(5²) + (6²)

To find the amount of roofing material required for a regular octagon, we use the formula for the area of a polygon: Area = 1/2 × Perimeter × Apothem. The perimeter of the octagon is 5 m × 8 sides, and the apothem is 6 m.

Explanation

To find the amount of roofing material required for a regular octagon, we use the formula for the area of a polygon: Area = 1/2 × Perimeter × Apothem. The perimeter of the octagon is 5 m × 8 sides, and the apothem is 6 m.

8) A city plaza is designed as a regular dodecagon (12-gon). The plaza is divided into equal sectors from the center. What is the measure of each central angle, which can be used to place benches symmetrically?

30°

45°

60°

90°

In a regular dodecagon, the total interior angle is 360°. To find the measure of each central angle, divide 360° by the number of sides (12): 360° / 12 = 30°. Thus, each central angle measures 30°.

Explanation

In a regular dodecagon, the total interior angle is 360°. To find the measure of each central angle, divide 360° by the number of sides (12): 360° / 12 = 30°. Thus, each central angle measures 30°.

9) A bridge support uses triangular braces modeled as equilateral triangles. This is chosen because:

Equilateral triangles tessellate:

Equilateral triangles maximize area:

Equilateral triangles have diagonals:

Equilateral triangles are rigid and stable:

Equilateral triangles are chosen for bridge supports because they provide rigidity and stability, which are essential for supporting weight and maintaining the structural integrity of the bridge.

Explanation

Equilateral triangles are chosen for bridge supports because they provide rigidity and stability, which are essential for supporting weight and maintaining the structural integrity of the bridge.

10) A designer makes a table with a regular hexagonal top. Each side is 2 ft. Trim costs $4 per foot. What will be the total cost of the trim?

$24

$36

$40

$48

To find the total cost of the trim, first calculate the perimeter of the hexagonal top. The perimeter is the sum of all sides, which is 6 sides times 2 ft each, giving 12 ft. The cost of the trim is then 12 ft times $4 per foot, totaling $48.

Explanation

To find the total cost of the trim, first calculate the perimeter of the hexagonal top. The perimeter is the sum of all sides, which is 6 sides times 2 ft each, giving 12 ft. The cost of the trim is then 12 ft times $4 per foot, totaling $48.

11) A sidewalk is paved with 50 equilateral triangular tiles, each with side length 3 m. The cost of paving is $12 per square meter. What is the approximate total cost?

$2100

$2200

$2340

$2500

To calculate the total cost, first find the area of one equilateral triangular tile using the formula (sqrt(3)/4) * side^2. For side length 3 m, the area is approximately 3.897 m². Then, multiply this area by 50 tiles to get the total area (approximately 194.85 m²). Finally, multiply the total area by the cost per square meter ($12) to find the total cost, which is approximately $2340.

Explanation

To calculate the total cost, first find the area of one equilateral triangular tile using the formula (sqrt(3)/4) * side^2. For side length 3 m, the area is approximately 3.897 m². Then, multiply this area by 50 tiles to get the total area (approximately 194.85 m²). Finally, multiply the total area by the cost per square meter ($12) to find the total cost, which is approximately $2340.

12) A company is designing a clock face in the shape of a regular 12-gon (dodecagon). Why is this shape effective for marking the hours evenly?

The central angles divide the circle evenly into 12 parts:

The diagonals are all congruent

The sides are perpendicular to each other

The polygon eliminates reflections

A regular dodecagon has 12 equal central angles, ensuring that each hour mark is evenly spaced around the clock face, providing clarity and uniformity in time representation.

Explanation

A regular dodecagon has 12 equal central angles, ensuring that each hour mark is evenly spaced around the clock face, providing clarity and uniformity in time representation.

13) A tabletop is shaped like a regular octagon. Which term best describes the line segment from the center of the octagon to one of its vertices?

Radius

Apothem

Diagonal

Altitude

The line segment from the center of a regular octagon to one of its vertices is called the radius. In a regular polygon, the radius connects the center to the corners (vertices) of the shape.

Explanation

The line segment from the center of a regular octagon to one of its vertices is called the radius. In a regular polygon, the radius connects the center to the corners (vertices) of the shape.

14) A student models a pizza as a regular 12-gon. Cutting it into 12 equal slices uses which property?

Each central angle = 30°

Each central angle = 45°

Each central angle = 60°

Each central angle = 90°

In a regular 12-gon, the total angle around the center is 360°. When divided into 12 equal slices, each slice has a central angle of 360°/12 = 30°.

Explanation

In a regular 12-gon, the total angle around the center is 360°. When divided into 12 equal slices, each slice has a central angle of 360°/12 = 30°.

15) Why are regular polygons often used in tiling patterns for floors and walls?

They minimize the number of diagonals:

Their equal sides and angles allow for uniform, gapless designs:

They prevent rotational symmetry:

They eliminate parallel lines:

Regular polygons have equal sides and angles, which makes it possible to fit them together without any gaps, resulting in a seamless and aesthetically pleasing design.

Explanation

Regular polygons have equal sides and angles, which makes it possible to fit them together without any gaps, resulting in a seamless and aesthetically pleasing design.

16) A fountain is shaped like a regular octagon with side length 4 m and apothem 4.8 m. Which formula gives the area of the fountain's surface?

1/2 × perimeter × apothem

1/2 × base × height

πr²

1/2 × diagonal

The area of a regular polygon can be calculated using the formula 1/2 × perimeter × apothem, where the perimeter is the total length of all sides, and the apothem is the distance from the center to the midpoint of a side.

Explanation

The area of a regular polygon can be calculated using the formula 1/2 × perimeter × apothem, where the perimeter is the total length of all sides, and the apothem is the distance from the center to the midpoint of a side.

17) A bridge support is reinforced with triangular frames. Engineers choose equilateral triangles instead of scalene triangles. Why is this choice more effective?

Equilateral triangles have all sides equal, so forces are evenly distributed along the frame:

Equilateral triangles have more diagonals than scalene triangles:

Equilateral triangles always contain a right angle for balance:

Equilateral triangles have unequal angles, which increases rigidity:

Equilateral triangles provide structural integrity because their equal sides and angles allow for uniform distribution of stress, making them a more stable choice in engineering applications compared to scalene triangles, which can lead to uneven force distribution.

Explanation

Equilateral triangles provide structural integrity because their equal sides and angles allow for uniform distribution of stress, making them a more stable choice in engineering applications compared to scalene triangles, which can lead to uneven force distribution.

18) A fountain is shaped as a regular pentagon. Its apothem is 6 m and the area is 75 m². What is the walking distance around it?

20 m

25 m

30 m

40 m

The walking distance around the fountain, also known as the perimeter, can be calculated using the area and apothem of the pentagon. The formula for the area of a regular polygon is Area = (Perimeter × Apothem) / 2. Rearranging this gives us Perimeter = (2 × Area) / Apothem. Plugging in the values gives us a perimeter of 25 m.

Explanation

The walking distance around the fountain, also known as the perimeter, can be calculated using the area and apothem of the pentagon. The formula for the area of a regular polygon is Area = (Perimeter × Apothem) / 2. Rearranging this gives us Perimeter = (2 × Area) / Apothem. Plugging in the values gives us a perimeter of 25 m.

19) A toy company makes spinning tops shaped like regular polygons. Which feature ensures balanced spinning?

Equal angles

Equal sides

Both equal sides and angles

Unequal diagonals

Balanced spinning in regular polygons is achieved when both the sides and angles are equal, ensuring uniform weight distribution and stability during rotation.

Explanation

Balanced spinning in regular polygons is achieved when both the sides and angles are equal, ensuring uniform weight distribution and stability during rotation.

20) A mural uses repeating squares and equilateral triangles. Why do these polygons fit together neatly?

Their interior angles are divisors of 360°:

Their sides are equal:

They have diagonals:

They have right angles:

Polygons fit together neatly when their interior angles can be combined to fill a 360° angle without gaps or overlaps. In this case, the interior angles of squares (90°) and equilateral triangles (60°) are divisors of 360°, allowing them to tessellate seamlessly.

Explanation

Polygons fit together neatly when their interior angles can be combined to fill a 360° angle without gaps or overlaps. In this case, the interior angles of squares (90°) and equilateral triangles (60°) are divisors of 360°, allowing them to tessellate seamlessly.