Area of a Rectangle, Triangle, Circle & Sector, Trapezoid, Square, Parallelogram, Rhombus, Geometry

The area of a rectangle is calculated by multiplying its length by its width. For example, a rectangle with a length of 8 and a width of 5 has an area of 40 square units. Remember to include units, such as square feet, if they are provided.

For a right triangle, the area is 1/2 * base * height. If the base is 10 and the height is 8, the area is (1/2) * 10 * 8 = 40 square units. For a general triangle, the base needs to be the entire length of the side to which the height is perpendicular. For an equilateral triangle with side 's', the area formula is (sqrt(3)/4) * s^2.

The area of a square is simply the side length squared (s^2). If a square has a side length of 9, its area is 9 * 9 = 81 square units.

The area of a circle is given by Pi * r^2, where 'r' is the radius. If the diameter is 10 cm, the radius is 5 cm, so the area is Pi * 5^2 = 25Pi square cm. To find the area of a sector of a circle, use the formula (angle in degrees / 360) * Pi * r^2. For a semicircle, the area is (1/2) * Pi * r^2.

The area of a parallelogram is calculated by multiplying its base by its height (base * height). If the base is 8 and the height is 12, the area is 8 * 12 = 96. If you're given a slant height, you may need to use the Pythagorean theorem to find the actual height.

The area of a trapezoid is 1/2 * (base1 + base2) * height. If the bases are 10 and 20, and the height is 8, the area is (1/2) * (10 + 20) * 8 = 120. For more complex trapezoids, you might need to break them into simpler shapes like rectangles and triangles to find the height.

The area of a rhombus is 1/2 * diagonal1 * diagonal2. For a rhombus with diagonals of 10 and 12, the area is (1/2) * 10 * 12 = 60. Remember that a rhombus has all sides congruent and its diagonals bisect each other at 90 degrees. This property can be used with the Pythagorean theorem to find missing diagonal lengths if given side lengths.

If you have two sides and the included angle (SAS) of a triangle, you can find its area using the formula: (1/2) * side1 * side2 * sin(included angle). For example, with sides 12 and 10 and an included angle of 30 degrees, the area is (1/2) * 12 * 10 * sin(30) = 30.

For a scalene triangle where all three side lengths (a, b, c) are known, you can use Heron's formula. First, calculate the semi-perimeter (s) which is (a + b + c) / 2. Then, the area is sqrt(s * (s - a) * (s - b) * (s - c)). For sides 9, 10, and 11, the semi-perimeter is 15, and the area is 30 * sqrt(2).

If the diagonal of a square is given, you can find the side length using the Pythagorean theorem (s^2 + s^2 = d^2) and then calculate the area (s^2). For a diagonal of 10 * sqrt(2), the side length is 10, and the area is 10^2 = 100.

To find the area of a shaded region when a triangle is inside a circle, subtract the area of the triangle from the area of the circle. If the triangle's base and height are equal to the circle's radius (e.g., in a right triangle where the center is a vertex), the shaded area is Pi * r^2 - (1/2) * base * height. For a radius of 8, this would be 64Pi - 32.