Finding the Area of Figures - Research Paper Example

Since the length of the base is b, there will b a number of square units (shaded region) in one row. Furthermore, as the height of the rectangle is h, there will h a number of rows with b number of square units in one row. Therefore, the total number of square units in the rectangle will be ‘b times h’, which is the area of the rectangle. Thus, the area of a reachable is given by:

The value of base b and h in figure 1 are 12 and 8, respectively. Therefore, the Area of the rectangle will be:

 Square-units

Square

A square is a special case of a rectangle in which both base and height are equal. Figure 2 shows a square with each side equal to b.

Figure 2: Square

The area of a square can be calculated the same as the area of a rectangle. The area of a square is given by:

The value of base b (or h) in figure 1 is 8. Therefore, the Area of the square will be:

 Square-units

Parallelogram

Figure 3 shows a parallelogram (enclosure ABCD) with base b and height h.

Figure 3: Parallelogram

Line DC is extended to point F. Line AE is perpendicular to line DC and line BF is perpendicular to line DF. The shape (triangle) represented by enclosures ADE and BCF are the same (congruent triangles). Therefore, if we cut part ADE from the parallelogram from the left and place this to the right on part BCF then the enclosure ABFE will be a rectangle with base b and height h. Therefore, the area of the parallelogram ABCD will be equal to the area of the rectangle ABFE that is given by:

Triangle

Figure 4 shows a triangle (enclosure ABC) with base b and height h.

Figure 4: Triangle

Line DA is parallel to line BC and line DB is parallel to line AC. The enclosure DACB represents a parallelogram with base b and height h.  The line AB divides the parallelogram DACB into two congruent triangles. Therefore, the area of the triangle ABC will be half the area of the parallelogram DACB, which is given by:

Trapezoid

Figure 5 shows a trapezoid (enclosure EFGH) with bases b1 and b2, and height h. This trapezoid can be separated into two triangles, triangle FGH, and triangle FEH. Thus, the area of the trapezoid will be the sum of these two triangles. The triangle FGH with base b1 and height h. and the triangle FEH with base b2 and height h.

Figure 5: Trapezoid

The area of the trapezoid will be given by:

Circle

Figure 5 shows a circle (left) with a radius of r.

Figure 6: Circle (Left)

The circumference C of a circle is equal to its diameter d times π, or 2 times its radius r times π. Finding the area of a circle is related to finding the area of a parallelogram. A circle can be separated into congruent wedge-like pieces, as shown in figure 6 (left). These wedge-like pieces can be arranged to form a figure similar to a parallelogram as shown in figure 6 (right). Thus, the circle has an area that is relatively close to the area of the parallelogram-shaped figure. Therefore, we can use the formula for the area of a parallelogram to find the area of a circle.

The parallelogram formed with congruent wedge-like pieces has a base equal to one-half the circumference and a height equal to the radius of the circle. Therefore, the area of the circle is given by:

Conclusion

In conclusion, the only formula that someone really needs to remember to find the area of any of the six figures a rectangle, a square, a parallelogram, a triangle, a trapezoid, and a circle is ‘Area = Base times Height’.