Pre-Calculus66 views·Updated Aug 13, 2026·2 pages

Trigonometry Basics: Understanding Polygons

Demi Zenit@demizenit_ftbu

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Circles and polygons have a special relationship when they're arranged...

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Inscribed and Circumscribed Polygons

When a polygon sits inside a circle with all its vertices touching the circle, it's called an inscribed polygon. The area of such a polygon can be found using the formula: $A = \frac{1}{2}nr^2 \sin\frac{2\pi}{n}$ , where r is the radius of the circle and n is the number of sides.

A circumscribed polygon wraps around a circle with all its sides touching the circle at exactly one point. For these polygons, the area formula is: $A = nr^2 \tan\frac{\pi}{n}$ , where r is the radius of the inscribed circle.

For triangles specifically, you can find the radius of a circumscribed circle using $r = \frac{abc}{4\sqrt{s(s-a)(s-b)(s-c)}}$ where s = $\frac{1}{2}(a+b+c)$ . Similarly, the radius of an inscribed circle in a triangle is $r = \frac{\sqrt{s(s-a)(s-b)(s-c)}}{s}$ .

Remember This! The sides of a regular polygon inscribed in a circle can be calculated using $a = 2r \sin\frac{\pi}{n}$ - this is super helpful when constructing or analyzing geometric shapes.

Circle measurements are also important to know: the circumference is $2πr$ or $πd$ , and the area is $πr^2$ or $\frac{1}{4}πd^2$ . The center of an inscribed circle is equidistant from all sides of the circumscribed polygon, while the apothem of the circumscribed polygon equals the radius of the inscribed circle.

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Regular Polygons and Their Properties

Regular polygons have equal sides and equal angles, making them perfectly symmetrical. The center of a regular polygon is a special point that serves as the center for both its inscribed and circumscribed circles.

A radius of a regular polygon connects the center to any vertex, and is also the radius of the circumscribed circle. The apothem is the perpendicular distance from the center to any side, and equals the radius of the inscribed circle. Central angles form between two radii drawn to consecutive vertices.

You can find the area of any regular polygon using the formula $\text{Area} = \frac{1}{2}ap$ , where a is the apothem and p is the perimeter. For example, a regular hexagon with an apothem of $5\sqrt{3}$ and a perimeter of 60 has an area of $150\sqrt{3}$ square units.

Cool Fact! The area of a circle circumscribed around a square ( $S_1$ ) is exactly twice the area of a circle inscribed within the same square ( $S_2$ ). This relationship shows the elegant proportions found in geometry.

The central angle in a regular polygon can be calculated by dividing 180° by the number of sides. This angle helps determine many other measurements, including the side length when given the radius.

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Pre-Calculus66 views·Updated Aug 13, 2026·2 pages

Trigonometry Basics: Understanding Polygons

Demi Zenit@demizenit_ftbu

Add to Google sources

Circles and polygons have a special relationship when they're arranged to share key points. This topic explores how regular polygons can be inscribed inside circles or circumscribe around them, along with the formulas needed to calculate their dimensions and areas.

of 2

Inscribed and Circumscribed Polygons

Remember This! The sides of a regular polygon inscribed in a circle can be calculated using $a = 2r \sin\frac{\pi}{n}$ - this is super helpful when constructing or analyzing geometric shapes.

of 2

Regular Polygons and Their Properties

Cool Fact! The area of a circle circumscribed around a square ( $S_1$ ) is exactly twice the area of a circle inscribed within the same square ( $S_2$ ). This relationship shows the elegant proportions found in geometry.

The central angle in a regular polygon can be calculated by dividing 180° by the number of sides. This angle helps determine many other measurements, including the side length when given the radius.

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Students love us — and so will you.

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Samantha KlichAndroid user

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Inscribed and Circumscribed Polygons

Regular Polygons and Their Properties

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