RÁPIDO e FÁCIL | POLÍGONOS REGULARES INSCRITOS NA CIRCUNFERÊNCIA

Summary

This video covers the concept of inscribed regular polygons, starting with equilateral triangles. It explains how to relate the radius of a circle to the side lengths of different polygons like squares and hexagons. The presenter demonstrates formulas for calculating the radius, area, and circumference of polygons inscribed within a circle, offering tips and shortcuts for solving related problems. Key concepts include understanding the relationship between the side of a polygon and the radius, the use of the apothem, and breaking down shapes like hexagons into simpler triangles for easier calculations.

Takeaways

• 😀 A polygon is said to be 'inscribed' when it fits perfectly inside a circle, with all vertices touching the circumference.
• 😀 An equilateral triangle inscribed in a circle has the radius of the circle equal to two-thirds of the height of the triangle.
• 😀 The area and circumference of a circle can be determined using the radius, with formulas: Circumference = 2πr and Area = πr².
• 😀 The side of an equilateral triangle inscribed in a circle can be related to the radius using the formula: Side = 2 × (Radius × √3) / 3.
• 😀 For a square inscribed in a circle, the diagonal of the square is equal to twice the radius of the circle.
• 😀 The radius of a circle can be calculated from the side length of an inscribed square using the relation: Radius = (Side × √2) / 2.
• 😀 The area of the circle can be calculated by subtracting the area of the inscribed square from the area of the circle itself.
• 😀 A regular hexagon inscribed in a circle can be divided into 6 equilateral triangles, where the side length of the hexagon is equal to the radius of the circle.
• 😀 The apothem of a regular polygon is the distance from the center to the midpoint of a side, and it can be calculated in various polygons such as triangles, squares, and hexagons.
• 😀 In an equilateral triangle, the apothem is one-third of the triangle's height, while in a hexagon, the apothem is equal to the height of the equilateral triangles that form it.

Q & A

What is the concept of a polygon inscribed in a circle?

A polygon is considered inscribed in a circle when all of its vertices lie on the circumference of the circle. The circle is referred to as the circumscribed circle.

What does 'regular' mean in the context of polygons?

'Regular' refers to a polygon where all sides and all angles are equal. For example, an equilateral triangle is a regular polygon because all its sides and angles are identical.

How is the radius of a circle related to an equilateral triangle inscribed in it?

The radius of the circle is two-thirds of the height of the equilateral triangle. This relationship can be used to find the radius when the side length of the triangle is known.

How can you find the area of the circle when the radius is given for an inscribed triangle?

To find the area of the circle, you use the formula π * r², where 'r' is the radius of the circle. The radius can be derived from the relationship between the side of the equilateral triangle and its height.

How is the diagonal of a square inscribed in a circle related to the radius?

The diagonal of the square is equal to the diameter of the circle, which is two times the radius. This relationship helps calculate the side length of the square from the radius.

What is the formula to calculate the area of a circle when the radius is known?

The formula for the area of a circle is A = π * r², where 'r' is the radius.

What is the significance of the apothem in polygons?

The apothem is the distance from the center of a polygon to the midpoint of one of its sides. It is important in calculating the area of regular polygons, especially when the radius is not directly involved.

What is the apothem of an equilateral triangle, and how is it calculated?

The apothem of an equilateral triangle is one-third of its height. The height can be calculated using the formula h = (side * √3) / 2, and the apothem is one-third of this value.

How is the area of a hexagon inscribed in a circle calculated?

The area of a hexagon inscribed in a circle can be calculated by dividing the hexagon into 6 equilateral triangles. The area of each triangle is (side² * √3) / 4, and the total area is 6 times that.

What is the relationship between the radius and the side length of a regular hexagon inscribed in a circle?

In a regular hexagon inscribed in a circle, the radius of the circle is equal to the side length of the hexagon. This is because each side of the hexagon is one of the radii of the circle.