Math Problem Statement

Mary designed a pendant. It is a regular octagon set in a circle. Suppose the opposite vertices are connected by line segments and meet at the center of the circle. What is the measure of each angle formed at the center?

Solution

The question involves a regular octagon inscribed in a circle and requires determining the angle at the center of the circle formed by opposite vertices. Let’s solve this step by step:

Step 1: Understand the Geometry

  • A regular octagon has 8 equal sides and is symmetrical.
  • The angle at the center is determined by the central angle subtended by opposite vertices.
  • The total angle around the circle is 360∘360^\circ.

Step 2: Central Angle Between Adjacent Vertices

  • The central angle between any two adjacent vertices of a regular octagon is: Central angle per segment=360∘8=45∘\text{Central angle per segment} = \frac{360^\circ}{8} = 45^\circ

Step 3: Opposite Vertices

  • To find the angle at the center formed by opposite vertices, we need to span half the circle. Therefore, the angle formed at the center by connecting opposite vertices is: Angle at center=45∘×4=180∘\text{Angle at center} = 45^\circ \times 4 = 180^\circ However, since the opposite vertices form pairs, the smaller angle between intersecting opposite segments is:
### Solution: 1. A **regular octagon** has 8 equal sides and vertices. 2. The total angle around the circle is $$360^\circ$$, and each central angle formed between two adjacent vertices is: \[ \text{Central angle between adjacent vertices} = \frac{360^\circ}{8} = 45^\circ.$$ 3. Opposite vertices are separated by half of the circle, which means the central angle they span is: $$45^\circ \times 4 = 180^\circ.$$ Thus, the smaller angles at the center, where all diagonals intersect, are determined by dividing $$360^\circ$$ by the number of segments formed at the center. Therefore, the angle formed between opposite diagonals at the center is: $$\boxed{45^\circ}.$$ The correct answer is **B: 45°**. Would you like a further breakdown or visual explanation? --- ### Related Questions: 1. How do we calculate the sum of interior angles of a regular octagon? 2. What is the relationship between the central angles and the vertices of a polygon? 3. How many diagonals can be drawn in a regular octagon? 4. What is the measure of each exterior angle of a regular octagon? 5. How do you determine angles in other regular polygons (e.g., hexagons)? **Tip:** Understanding symmetry in regular polygons greatly simplifies angle-related problems.

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Math Problem Analysis

Mathematical Concepts

Geometry
Polygons
Central Angles

Formulas

Central angle formula: 360° / number of sides

Theorems

Properties of regular polygons
Sum of angles around a point

Suitable Grade Level

Grades 9-12

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