In this section of our PERT Study Guide, we will cover geometry. The geometrical concepts, formulas, and calculations you need to know for the PERT—and in further mathematical studies—are covered below.
Perimeter
Perimeter is the total distance around a two-dimensional shape. Calculating the perimeter is essential in scenarios where you need to enclose a space or measure the total length of a shape’s edges. The formula for perimeter depends on the type of shape.
For a rectangle, the formula $P = 2(l + w)$ reflects the sum of the lengths of all four sides, where the opposite sides are equal. This formula is especially useful for tasks like fencing a rectangular yard or framing a picture.
For a square, the perimeter is $4 ×$ the length of one side, since all sides are the same.
For a triangle, the perimeter is simply the sum of the lengths of its three sides. This approach is versatile and applies to any triangle type—whether scalene, isosceles, or equilateral.
When in doubt, simply add up the length of all sides of a shape to get your answer.
Example: There is a garden that is 20 meters long and 10 meters wide. If you need to install a fence around it, how much fencing will you need?
$P = 2(20 + 10)$
$= 2 × 30 = 60 \text{ meters}$
Area
Area measures the space a shape covers, which is crucial for tasks like calculating the amount of paint required to cover a wall or the amount of carpet needed for a floor.
The area of a rectangle, $A = l × w$, multiplies its length by its width, offering a straightforward way to calculate space.
The area of a triangle, $A = \frac{1}{2} × \text{base} × \text{height}$, requires identifying the height perpendicular to the base, which can be less intuitive.
Example: There is a triangular patio with a base of 12 feet and a height of 8 feet. What is the area of the patio?
$A = \frac{1}{2} × \text{base} × \text{height}$
$= \frac{1}{2} × 12 × 8$
$= \frac{1}{2} × 96= 48 \text{ feet}^2$
Note that area is represented in units$^2$ while perimeter is simply represented in units.
Volume
Volume measures the space inside a three-dimensional object, making it critical for tasks like determining how much water a tank can hold or how much soil is needed to fill a garden bed. The formula for volume varies by shape.
For a rectangular prism, the volume is calculated by multiplying its three dimensions: $V = l × w × h$.
For a cylinder, the base area ($πr^2$) is multiplied by the height to account for the depth of the shape: $V = πr^2h$.
Example: You are filling a rectangular box that is 1 foot long, 3 feet wide, and 2 feet high. What is the box’s volume?
$V = 4 × 3 × 2 = 24 \text{ feet}^3$
Remember that area is calculating a 2-D space, so it is represented in units$^2$, and volume calculates a 3-D space, so it is represented in units$^3$.
Pythagorean Theorem
The Pythagorean Theorem, $a^2+ b^2 = c^2$, applies to right triangles and establishes a relationship between the triangle’s legs and its hypotenuse. It is fundamental in geometry and often serves as the basis for solving real-world problems involving right angles.
The legs of a triangle are the two sides that form the right angle, and the hypotenuse is the longest side, or the side across from the right angle.
To solve a right triangle, identify the two legs ($a$ and $b$) and the hypotenuse ($c$). Depending on the given values, the theorem helps find the missing side.
Example: Suppose a ladder leans against a wall, forming a right triangle. The ladder is 13 feet long and the base of the ladder is 5 feet away from the wall. Find the height of the wall.

$a^2 + b^2 = c^2$
$5^2 + b^2 = 13^2$
$25 + b^2 = 169$
$b^2 = 144$
$b = \sqrt{144} = 12 \text{ feet}$
Key Tip: When working with geometric problems, start by drawing a diagram to visualize the shape and label all known dimensions. This helps clarify relationships and ensures formulas are applied correctly.