Areas of Rectangles and Squares
Consider the rectangle in the following figure, where "b" is the base, and "h" is the height:
The area of this rectangle is given by the formula: $A_r=bh$
A square is just a rectangle in which $h=b$. Therefore, the area of a square is given by the formula: $A_s=bh=bb=b^2$
Area of Parallelograms
Consider the parallelogram EFGH (figure below), where "b" is the base, and "h" is the height. Let's split this parallelogram in three parts, with areas $A_1$ (rectangle), $A_2$ and $A_3$ (two right triangles):
If we move triangle $A_2$ to the side of triangle $A_3$, the new figure is a rectangle with base "b" and height "h":
Therefore, the area of the parallelogram is $A_p=bh$, that is, the same formula used to compute the area of a rectangle.
Attention: "h" is the height of the parallelogram, not the size of its side.
Area of Triangles
Consider triangle ABC (figure below). Triangle ACD has the same side sizes and angles as triangle ABC, but is rotated 180 degrees. As we place both triangles side by side, they form the parallelogram ABCD, with base "b" and height "h":
Since triangles ABC and ACD are congruent, they have the same area (half the area of the parallelogram ABCD). Therefore the area of a triangle is given by the formula:
$A_t=\frac{bh}{2}$
Heron's Formula
As seen in chapter General Triangles, the height of any triangle is given by the formula:
$h_a=\frac{2}{a}\sqrt{p(p-a)(p-b)(p-c)}$,
where "p" is half of the triangle's perimeter, "a", "b" and "c" are the sizes of the triangle's sides, and side "a" is the base of the triangle.
The area of triangles is given by the formula:
$A_t=\frac{bh}{2}$. Thus,
$A_t=\frac{a\frac{2}{a}\sqrt{p(p-a)(p-b)(p-c)}}{2}$
$A_t=\sqrt{p(p-a)(p-b)(p-c)}$
With this formula it is possible to compute the area of any triangle using just the size of its sides.
Area of Trapezoids
Consider trapezoid CDEF (figure below), with bases "B" and "b", and height "h":
Diagonal CE splits the trapezoid into two triangles with areas $A_1$ and $A_2$. The area of the trapezoid is the sum of these two areas:
$A_{trap}=A_1+A_2$
$A_{trap}=\frac{bh}{2}+\frac{Bh}{2}=\frac{(b+B)h}{2}$
Area of Rhombuses
Consider the rhombus in the following figure, divided by its two diagonals "D" e "d":
In parallelograms diagonals intersect each other in the midpoint. And in rhombuses, a particular case of parallelogram, diagonals also form a 90º angle in the intersection.
Diagonal "D" connects vertices A and B, spliting the rhombus into two triangles with height "d/2". The sum of the areas of these two triangles is the area of the rhombus:
$A_L=\frac{Dd/2}{2}+\frac{Dd/2}{2}$
$A_L=\frac{Dd}{2}$
Area of Quadrilaterals using Diagonals
Consider quadrilateral ABCD in the following figure, where the two diagonals intersect forming an angle "a":
The dotted red lines are parallel to the diagonals, forming parallelogram EFGH, whose sides measure the same as the diagonals ($d_1$ and $d_2$). The area of this parallelogram can be computed using the formula:
$A_{par}=d_1d_2sin{(a)}$,
where $d_2sin{(a)}$ is the height of the parallelogram, when we consider $d_1$ as its base.
Note that parallelogram EFGH is formed by two congruent triangles denoted as "1", two congruent triangles denoted as "2", two congruent triangles denoted as "3", and two congruent triangles denoted as "4". Therefore, the area of quadrilateral ABCD is half the area of parallelogram EFGH:
$A_{quadrilateral}=\frac{d_1d_2sin{(a)}}{2}$
Area of Circular Sectors
The area of a circle is given by the formula $A_C=\pi{r}^2$, where $r$ is the radius.
A circular sector is the portion of a circle enclosed in an agle $\alpha$. The area of the sector is given by the formula:
$A_{\alpha}=(\frac{\alpha}{360º})\pi{r}^2$, when $\alpha$ is measured in degrees.
or
$A_{\alpha}=(\frac{\alpha}{2.\pi}).\pi.r^2$, when $\alpha$ is measured in radians.
Area of an Annulus
The area of an annulus (following figure) is the difference between the area of the larger circle and the area of the smaller circle:
$A_{Annulus}=\pi{R}^2-\pi{r}^2=\pi(R^2-r^2)$
Solved SAT Practice Tests
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SAT Practice Tests - Areas of Geometrical Figures and
Additional Practice Tests - Areas of Geometrical Figures
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