Finding the area of various shapes, including triangles, circles, and parallelograms, may seem overwhelming, but with the right formulas, it can be a straightforward process. In this article, we will delve into easy ways to calculate the area of these shapes using different approaches.

To determine the area of a triangle, you can use either of two methods - using lengths or applying trigonometry.

The formula for finding the area of a triangle is **1/2 x base x height**. Here, the base (b) refers to the length of the triangle's base, and the height (h) refers to the length of a line drawn perpendicular to the base.

With the knowledge of two lengths and the angle between them, you can use trigonometry (SAS - side, angle, side) to calculate the area of a triangle. The formula for this is **1/2 x a x b x sin(C)**, where a and b are the known lengths, and sin(C) is the sine of the angle between them.

- Determine the area of a triangle with a base length of 7 cm and a height of 50 cm.
- Find the base length and height of a triangle with an area of 9cm², knowing that the height is twice the length of the base.

The formula for finding the area of a circle is **π x (radius)²**. Here, π (pi) is an irrational number (approximately 3.14) that relates the circumference of a circle to its diameter. You can also determine the area of a sector (a fraction of a circle) by taking a fraction of the full area using the formula **(angle/360) x π x (radius)²**.

- Determine the area of a circle with a radius of 3 cm, giving your answer in exact form.
- Find the area of a sector with an angle of 34° and a radius of 2 km, giving your answer to two decimal places.

The formula for determining the area of a parallelogram is **base x perpendicular height** (b x h). This formula is closely related to that of a triangle, as a parallelogram can be split into two identical triangles when divided by its diagonal.

The area of a parallelogram can also be calculated using the trigonometric method for finding the area of a triangle: **(a x b x sin(C)) x 2**, where a and b are adjacent lengths, and C is the angle between them.

If a parallelogram has a base that is three times its height, and the area is 108m², what is the length of the base?

Let the height = x. Substituting known values into the formula, we get:

**108 = 3x x x** (since the base is three times the height)

Hence, x = 6. Therefore, the base length is **18 meters**.

When dealing with shapes that are not commonly encountered, such as a rectangle with semicircles attached to the ends or a regular polygon with an unknown number of sides, it is necessary to break them down into smaller, more familiar shapes in order to calculate their areas. For example, a rectangle with semicircles attached to the ends can be divided into a rectangle and two semicircles, while a regular polygon can be split into equally sized isosceles triangles.

Let's take the Australian 50 cent coin as an example. This coin has the shape of a regular dodecagon, which has 12 sides. If eight of these coins can fit on an Australian $5 note, what fraction of the note remains uncovered? As there is no direct formula for calculating the area of a dodecagon, it can be divided into 12 isosceles triangles with a vertex at the center. Each triangle has two sides of equal length (the radius of the dodecagon) and an angle of 30 degrees between them. Using the formula for the area of a triangle, A = 1/2 * base * height, the area of each triangle can be calculated in terms of the radius. And since there are eight dodecagons, the total area they cover is 8 * 12 * 1/2 * r * r * sin30 degrees. Similarly, the area of the note can be calculated, knowing that two coins fit vertically and four fit horizontally, giving it a height of 4r and a length of 85. This gives us the equation A = 4r * 85 = 340r. To find the fraction of the note that is not covered, the total area covered by the coins can be subtracted from the total area of the note, giving us (340r - 8 * 12 * 1/2 * r * r * sin30 degrees)/340r = 2/17, or approximately 11.76%.

**Important Concepts to Remember:**- The area of a circle can be found using A = (pi) * r², where r is the radius
- There are two formulas for calculating the area of a triangle: A = 1/2 * base * height or A = 1/2 * a * b * sinC, where a and b are adjacent sides and C is the angle between them
- The area of a parallelogram is calculated using A = base * height, or A = a * b * sinC, where a and b are adjacent sides and C is the angle between them
- The area of a rectangle can be found using A = length * width
- The area of a square is A = side * side, or A = a², where a is the length of any side of the square

In some instances, shapes may consist of multiple smaller shapes, and it is crucial to identify how they are put together in order to accurately calculate their areas.

Now, let's put our knowledge into practice with some examples!

**Question 1:** How do you find the area of a circle?**Answer:** You can use the formula A = (pi) * r², where r is the radius of the circle.**Question 2:** How do you find the area of a triangle?**Answer:** There are two main formulas for calculating the area of a triangle: A = 1/2 * base * height, or A = 1/2 * a * b * sinC, where a and b are adjacent sides and C is the angle between them.**Question 3:** How do you find the area of a square?**Answer:** Since all sides are equal, you can use the formula A = side * side, or A = a², where a is the length of any side of the square.

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