Triangle Area and Circle Radii Formulas

This page continues with formulas for various triangle types and introduces the general area formula for any triangle.

For an equilateral triangle, the following formulas are provided:

• Radius of the circumscribed circle: R = a / √3, where 'a' is the side length
• Radius of the inscribed circle: r = a / (2√3)

Highlight: In an equilateral triangle, the ratio of the circumscribed circle radius to the inscribed circle radius is always 2:1.

The page also presents the general formula for calculating the area of any triangle using the semi-perimeter:

P = √$s(s-a)(s-b)(s-c)$

Where:

• P is the area of the triangle
• s is the semi-perimeter: s = $a + b + c$ / 2
• a, b, and c are the lengths of the triangle's sides

Example: For a triangle with sides 3, 4, and 5, the semi-perimeter s = $3 + 4 + 5$ / 2 = 6. The area would be P = √$6(6-3)(6-4)(6-5)$ = √$6 * 3 * 2 * 1$ = √36 = 6.

The radius of the circumscribed circle for any triangle can be calculated using:

R = abc / (4P)

Where abc is the product of the three side lengths, and P is the area of the triangle.

Definition: The circumscribed circle, also known as the circumcircle, is the unique circle that passes through all three vertices of a triangle.

These formulas provide powerful tools for analyzing triangles and their associated circles, allowing for the calculation of various geometric properties based on known triangle dimensions.

Inscribed and Circumscribed Circles of a Triangle

This page introduces the concepts of okrąg wpisany w trójkącie (inscribed circle) and okrąg opisany na trójkącie (circumscribed circle) for different types of triangles. It provides formulas for calculating the radii of these circles and related triangle properties.

Definition: The inscribed circle of a triangle is the largest circle that can be drawn inside the triangle, touching all three sides. The circumscribed circle passes through all three vertices of the triangle.

Highlight: The center of the inscribed circle is located at the intersection of the angle bisectors of the triangle.

For a trójkąt prostokątny (right triangle), the following formulas are provided:

• Area: P = ah/2, where 'a' is the base and 'h' is the height
• Radius of the circumscribed circle: R = c/2, where 'c' is the hypotenuse

Example: In a right triangle with legs 3 and 4, and hypotenuse 5, the radius of the circumscribed circle would be R = 5/2 = 2.5.

For a general triangle, the radius of the inscribed circle is given by:

r = P / s, where P is the area and s is the semi-perimeter

The semi-perimeter is calculated as: s = $a + b + c$ / 2, where a, b, and c are the side lengths of the triangle.

For a trójkąt równoramienny (isosceles triangle), specific formulas are provided for the radius of the circumscribed circle and the height:

• R = a² / (4h), where 'a' is the length of the equal sides and 'h' is the height
• h = a√3 / 2, relating the height to the side length

Vocabulary: The height $h$ in an isosceles triangle is the perpendicular line segment from the vertex angle to the base.