Rectangle Area Calculator

Tags: Area of 2D Shapes

The calculator allows you to calculate the area of a rectangle, using the lengths of its sides, the length of the diagonal and the angle between the diagonal and the side, and other initial data. The formula for calculating the area through the sides is: \(S = a \times b\), where a and b are the lengths of the rectangle's sides.

The rectangle, one of the most basic geometric figures, serves as a key object in the study of area. This shape, characterized by a pair of parallel and equal sides, represents an ideal example for explaining the concept of area, as it can be easily calculated and represented as a model in the real world.

Definition and Fundamental Properties of a Rectangle

A rectangle is a quadrilateral with all right angles, that is, each angle equals 90 degrees. This definition emphasizes one of the rectangle's key properties, making it an important object in geometry and applied fields. The area of a rectangle is calculated by the formula \(S = a \times b\), where \(a\) and \(b\) are the lengths of the rectangle's sides.

The fundamental properties of a rectangle include:

• All angles equal 90 degrees, making it a rectangle.
• Opposite sides are equal and parallel to each other.
• The diagonals of a rectangle are equal and intersect, dividing each other in half.
• A rectangle can be considered a special case of a parallelogram, with at least one right angle.

These fundamental properties underlie the geometric characteristics of a rectangle and determine the methods for calculating its area and other parameters, as well as allow a deeper exploration and application of concepts related to rectangles in various mathematical and applied tasks.

Rectangle Area Formulas

The area of a rectangle is a measure of space within a quadrilateral with right angles. There are several ways to calculate the area of a rectangle depending on the known initial data.

Two Sides

If the lengths of two sides of the rectangle (\(a\) and \(b\)) are known, the area is found using the formula: \(S = a \times b\)

Diagonals and the Angle Between Them

With the known length of the diagonals (\(d\)) and the angle between them (\(\theta\)), the area is calculated as: \(S = \frac{1}{2} \times d^2 \times \sin(\theta)\)

Side and Diagonal

If the length of one side (\(a\)) and the diagonal (\(d\)) are known, the second side can be found using Pythagoras' theorem (\(b = \sqrt{d^2 - a^2}\)), and then the area: \(S = a \times \sqrt{d^2 - a^2}\)

Side and Diameter of the Circumscribed Circle

The diameter of the circumscribed circle (\(D\)) is related to the length of the diagonal of the rectangle, as the diagonal of the rectangle equals the diameter of the circumscribed circle. Knowing one side (\(a\)) and \(D\), the area can be found as: \(S = a \times \sqrt{D^2 - a^2}\)

Side and Perimeter

The perimeter of the rectangle (\(P\)) and one of the sides (\(a\)) allow expressing the second side (\(b\)) as \(b = \frac{P}{2} - a\), after which the area is found through: \(S = a \times \left(\frac{P}{2} - a\right)\)

Radius of the Circumscribed Circle and the Angle Between Diagonals

The area of the rectangle can also be found through the radius of the circumscribed circle (\(R\)) and the angle between the diagonals (\(\theta\)), which is equal to the doubled angle between the side of the rectangle and the diagonal: \(S = 2R^2 \sin(\theta)\)

These formulas demonstrate the variety of methods for calculating the area of a rectangle, making them applicable in a wide range of tasks, from academic to applied.

Examples of Calculating the Area of a Rectangle

For a clear understanding of the methods of calculating the area of a rectangle, let's consider several typical tasks:

• Calculating the area through two sides
  Let the lengths of the sides of the rectangle be 4 m and 6 m. Then the area of the rectangle \(S\) will be equal to \(S = a \times b = 4 \times 6 = 24\) sq.m.
• Calculating the area through the diagonal and the angle between diagonals
  If the length of the diagonal of the rectangle is 10 m, and the angle between the diagonals is 90 degrees (characteristic for a rectangle), the area can be found using the formula \(S = \frac{1}{2} \times d^2 \times \sin(\theta) = \frac{1}{2} \times 10^2 \times \sin(90^\circ) = 50\) sq.m.
• Calculating the area through a side and diagonal
  For a rectangle with a side of 3 m and a diagonal of 5 m, the second side will be equal to \(b = \sqrt{d^2 - a^2} = \sqrt{5^2 - 3^2} = 4\) m, and the area \(S = a \times b = 3 \times 4 = 12\) sq.m.
• Calculating the area through a side and the diameter of the circumscribed circle
  If a side of the rectangle is 6 m, and the diameter of the circumscribed circle is 10 m (equal to the length of the diagonal), then the second side is found as \(b = \sqrt{D^2 - a^2} = \sqrt{10^2 - 6^2} = 8\) m, therefore, \(S = 6 \times 8 = 48\) sq.m.
• Calculating the area through a side and perimeter
  If one side of the rectangle is 7 m, and its perimeter is 26 m, then the second side will be \(b = \frac{P}{2} - a = \frac{26}{2} - 7 = 6\) m, and the area \(S = 7 \times 6 = 42\) sq.m.
• Calculating the area through the radius of the circumscribed circle and the angle between diagonals
  For a rectangle with the radius of the circumscribed circle 5 m and the angle between the diagonals 90 degrees, the area is calculated as \(S = 2R^2 \sin(\theta) = 2 \times 5^2 \times \sin(90^\circ) = 50\) sq.m.

These examples demonstrate how different initial data can be used to calculate the area of a rectangle, providing useful skills for solving a wide range of tasks.

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