Area of Polygons

Introduction to Geometry and Measurement

By the end of this lesson, you will be able to calculate perimeter and area of common polygons, find the circumference and area of circles, and compute surface area and volume of 3D figures.

Perimeter

Perimeter: the total distance around a figure.

Rectangle: $P = 2l + 2w$ or $P = 2(l + w)$

Square: $P = 4s$

Triangle: $P = a + b + c$ (sum of all sides)

Regular polygon (all sides equal): $P = ns$ where $n$ = number of sides, $s$ = side length

Example: Find the perimeter of a rectangle with length 12 cm and width 7 cm.

$P = 2(12) + 2(7) = 24 + 14 = 38$ cm

Area of Polygons

Area: the measure of the interior of a 2D figure, measured in square units.

Rectangle: $A = l \times w$

Square: $A = s^2$

Triangle: $A = \dfrac{1}{2} \times b \times h$ (base × height, where height is perpendicular to the base)

Parallelogram: $A = b \times h$

Trapezoid: $A = \dfrac{1}{2}(b_1 + b_2) \times h$ ($b_1$ and $b_2$ are the two parallel bases)

Example: Find the area of a triangle with base 10 m and height 6 m.

$A = \dfrac{1}{2}(10)(6) = 30$ m²

Example: Find the area of a trapezoid with bases 8 ft and 12 ft, and height 5 ft.

$A = \dfrac{1}{2}(8 + 12)(5) = \dfrac{1}{2}(20)(5) = 50$ ft²

Circles

Circle terminology:

• Radius ($r$): distance from center to any point on the circle
• Diameter ($d$): distance across the circle through the center; $d = 2r$
• $\pi \approx 3.14159...$ (irrational number; use 3.14 or $\dfrac{22}{7}$ for approximation)

Circumference (perimeter of a circle):

$$C = 2\pi r = \pi d$$

Area of a circle:

$$A = \pi r^2$$

Example: Find the circumference and area of a circle with radius 5 cm.

$C = 2\pi(5) = 10\pi \approx 31.4$ cm

$A = \pi(5)^2 = 25\pi \approx 78.5$ cm²

Pythagorean Theorem

For a right triangle with legs $a$ and $b$ and hypotenuse $c$:

$$a^2 + b^2 = c^2$$