What are the formulas for calculating the perimeter and area of triangles, quadrilaterals, and circles?
Understand the Problem
The user is asking for the formulas to calculate the perimeter and area of common polygons. This includes 3-sided polygons (triangles), 4-sided polygons (quadrilaterals), and circles.
Answer
Triangle: $P = a + b + c$, $A = \frac{1}{2}bh$ or $A = \sqrt{s(s-a)(s-b)(s-c)}$ Square: $P = 4a$, $A = a^2$ Rectangle: $P = 2(l+w)$, $A = lw$ Circle: $C = 2\pi r$, $A = \pi r^2$
Answer for screen readers
Triangle:
- Perimeter: $P = a + b + c$
- Area: $A = \frac{1}{2}bh$ or $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$
Square:
- Perimeter: $P = 4a$
- Area: $A = a^2$
Rectangle:
- Perimeter: $P = 2(l+w)$
- Area: $A = lw$
Circle:
- Circumference: $C = 2\pi r$
- Area: $A = \pi r^2$
Steps to Solve
- Triangle - Perimeter
The perimeter of a triangle is the sum of the lengths of its three sides. If the sides are $a$, $b$, and $c$, then the perimeter $P$ is:
$P = a + b + c$
- Triangle - Area
The area of a triangle can be calculated in several ways, depending on the information available. Here are two common formulas:
-
Using base $b$ and height $h$:
$A = \frac{1}{2}bh$
-
Using Heron's formula, where $s$ is the semi-perimeter $s = \frac{a+b+c}{2}$:
$A = \sqrt{s(s-a)(s-b)(s-c)}$
- Quadrilateral - Perimeter
For a general quadrilateral with sides $a$, $b$, $c$, and $d$, the perimeter $P$ is the sum of the lengths of all four sides:
$P = a + b + c + d$
- Square - Area and Perimeter
For a square with side length $a$, the perimeter $P$ and area $A$ are:
$P = 4a$
$A = a^2$
- Rectangle - Area and Perimeter
For a rectangle with length $l$ and width $w$, the perimeter $P$ and area $A$ are:
$P = 2l + 2w = 2(l+w)$
$A = lw$
- Circle - Circumference (Perimeter)
The perimeter of a circle is called the circumference (C). If the radius of the circle is $r$, then the circumference is:
$C = 2\pi r$
- Circle - Area
The area $A$ of a circle with radius $r$ is:
$A = \pi r^2$
Triangle:
- Perimeter: $P = a + b + c$
- Area: $A = \frac{1}{2}bh$ or $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \frac{a+b+c}{2}$
Square:
- Perimeter: $P = 4a$
- Area: $A = a^2$
Rectangle:
- Perimeter: $P = 2(l+w)$
- Area: $A = lw$
Circle:
- Circumference: $C = 2\pi r$
- Area: $A = \pi r^2$
More Information
- $\pi$ (pi) is approximately equal to 3.14159.
- Heron's formula is useful when you know the lengths of all three sides of a triangle but not the height.
Tips
- Forgetting to use the correct units. Make sure all measurements are in the same units before calculating area or perimeter.
- Using the diameter instead of the radius when calculating the circumference or area of a circle. Remember that the radius is half the diameter ($r = \frac{d}{2}$).
- Confusing perimeter and area. Perimeter is the distance around a shape, while area is the amount of surface it covers.
AI-generated content may contain errors. Please verify critical information
