Properties of Plane Figures

This page provides a comprehensive overview of various plane figures and their properties, focusing on triangles and quadrilaterals. The information is presented visually with diagrams and formulas for easy understanding.

Right-angled Triangle

The page begins with a right-angled triangle, showcasing the Pythagorean theorem.

Definition: The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse $c$ is equal to the sum of squares of the other two sides (a and b).

Formula: a² + b² = c²

This formula is crucial for calculating the długości boków w trójkącie prostokątnym (lengths of sides in a right-angled triangle).

Highlight: The page also includes a formula for calculating the wysokość w trójkącie prostokątnym (height in a right-angled triangle): h = $a * b$ / c

Equilateral Triangle

Next, the properties of an equilateral triangle are presented.

Formula: The height $h$ of an equilateral triangle is given by h = a√3 / 2, where 'a' is the length of a side.

Example: The area (P) of an equilateral triangle can be calculated using P = a²√3 / 4

Square

The properties of a square are illustrated, emphasizing its equal sides and right angles.

Definition: A square is a regular quadrilateral with four equal sides and four right angles.

Formula: The area of a square is given by P = a², where 'a' is the length of a side.

Highlight: The diagonal of a square is represented as a√2, which is an important property when considering przekątne kwadratu (diagonals of a square).

Rectangle

The rectangle's properties are shown next to the square for comparison.

Formula: The area of a rectangle is calculated as P = a * b, where 'a' and 'b' are the lengths of the sides.

Rhombus

The rhombus is presented with its diagonals and area formula.

Vocabulary: A rhombus is a quadrilateral with four equal sides.

Formula: The area of a rhombus is given by P = $e * f$ / 2, where 'e' and 'f' are the lengths of the diagonals.

Trapezoid

The trapezoid (or trapezium) is shown with its area formula.

Formula: The wzór na pole trapezu (formula for the area of a trapezoid) is P = $a + b$ * h / 2, where 'a' and 'b' are the lengths of the parallel sides and 'h' is the height.

Parallelogram

Finally, the parallelogram is presented with its area formula and angle relationships.

Formula: The area of a parallelogram is calculated as P = a * h, where 'a' is the base and 'h' is the height.

Highlight: The angles in a parallelogram have the following relationships: α + β = 180° and β + γ = 180°

This comprehensive guide serves as an excellent resource for students studying geometry, providing key formulas and visual representations for various plane figures.