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:

As the number of sides increases, a regular polygon becomes almost identical to a circle.

Explanation

As the number of sides increases, a regular polygon becomes almost identical to 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

A regular octagon has all sides and all angles equal, giving it perfect balance and symmetry — easy to recognize from any direction.

Explanation

A regular octagon has all sides and all angles equal, giving it perfect balance and symmetry — easy to recognize from any direction.

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

The hexagon shape covers space efficiently with the least material — that’s why bees use it to store honey with less wax!

Explanation

The hexagon shape covers space efficiently with the least material — that’s why bees use it to store honey with less wax!

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

Six hexagons fit perfectly around a point (6 × 120° = 720°), filling space without gaps or overlaps.

Explanation

Six hexagons fit perfectly around a point (6 × 120° = 720°), filling space without gaps or overlaps.

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

In an equilateral triangle, all sides and angles are equal, giving it perfect symmetry — easy for drivers to identify.

Explanation

In an equilateral triangle, all sides and angles are equal, giving it perfect symmetry — easy for drivers to identify.

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)

Use the area formula for a regular polygon:

Area = ½ × perimeter × apothem → ½ × (8 × 5) × 6.

Explanation

Use the area formula for a regular polygon:

Area = ½ × perimeter × apothem → ½ × (8 × 5) × 6.

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°

Divide 360° by the number of sides: 360 ÷ 12 = 30°. Each section of the plaza spreads evenly around the center.

Explanation

Divide 360° by the number of sides: 360 ÷ 12 = 30°. Each section of the plaza spreads evenly around the center.

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 don’t change shape easily, making them ideal for strength and balance in structures.

Explanation

Equilateral triangles don’t change shape easily, making them ideal for strength and balance in structures.

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

Perimeter = 6 × 2 = 12 ft.

Trim cost = 12 × $4 = $24.

Explanation

Perimeter = 6 × 2 = 12 ft.

Trim cost = 12 × $4 = $24.

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. Area of one triangle ≈ 3.9 m²,

Total area ≈ 3.9 × 50 = 195 m².

Cost = 195 × $12 = $2340.

Explanation

To calculate the total cost, first find the area of one equilateral triangular tile using the formula. Area of one triangle ≈ 3.9 m²,

Total area ≈ 3.9 × 50 = 195 m².

Cost = 195 × $12 = $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: Perimeter = 8 × 4 = 32 m.

So, Area = ½ × 32 × 4.8 = 76.8 m².

Explanation

The area of a regular polygon can be calculated using the formula: Perimeter = 8 × 4 = 32 m.

So, Area = ½ × 32 × 4.8 = 76.8 m².

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

Use the area formula: 75 = ½ × Perimeter × 6

→ Perimeter = (75 × 2) ÷ 6 = 25 m.

Explanation

Use the area formula: 75 = ½ × Perimeter × 6

→ Perimeter = (75 × 2) ÷ 6 = 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

Equal sides and angles mean the weight is evenly distributed around the center, so the top spins smoothly.

Explanation

Equal sides and angles mean the weight is evenly distributed around the center, so the top spins smoothly.

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:

The interior angles (90° for squares and 60° for triangles) fit exactly into 360°, allowing perfect tiling.

Explanation

The interior angles (90° for squares and 60° for triangles) fit exactly into 360°, allowing perfect tiling.