The English word perimeter comes from the Greek words peri, which means surrounding, and metron, which means measure. The perimeter of a two-dimensional shape or a one-dimensional length forms a closed path. The circumference of an ellipse or circle is its perimeter.
How is it calculated?
The perimeter of a triangle is calculated by summing the lengths of its sides. This method can be used for anything from measuring the thread needed to mark the perimeter of a soccer field to estimating the expense of fencing in your backyard garden.
Triangle's perimeter
Any two-dimensional figure's perimeter is determined by its surroundings. The perimeter of any closed object can be computed by summing the lengths of its sides. It is equal to the total of a polygon's sides.
For example, a regular polygon's perimeter equals the sum of its sides times the number of sides.
A polygon with three sides is called a triangle. The lengths of all three sides of a triangle are added up to find its perimeter, the total length of time travelled along its edge.
The perimeter is given in m, cm, km, etc., as it measures the lengths.Â
In real life, we frequently apply the concept of perimeter. For instance, we measure the perimeter of the backyard to determine how much wire we need to install lights around the home or a fence.
FormulaÂ
For a triangle’s perimeter, the formula is:
P= a + b + c
where a, b, and c are the triangles' sides. The perimeter of any two-dimensional figure is its circumference. For instance, the perimeter of a circle is its circumference, i.e., 2Ï€r
There are various techniques for figuring out the perimeter of several types of triangles
The Equilateral Triangle's Perimeter
An equilateral triangle is a triangle in geometry where each of the three sides is the same length. An equilateral triangle's perimeter can also be determined using the following formula since all of its sides are identical to one another:
Perimeter, P = 3L
where L is the triangle's side length.
This formula is simply derived from the primary perimeter of the triangle formula.
As seen above, any triangle's perimeter equals the sum of its sides. On the other hand, every side in an equilateral triangle is the same length.
A= B= C= L.
Therefore, the perimeter of the equilateral triangle is equal to A + B + C = L + L + L = 3L.
Perimeter of an Isosceles Triangle
An isosceles triangle is defined in geometry as having at least two equal-length sides. The perimeter of an isosceles triangle can also be computed using the following formula since two of its sides are equal, and one side is longer than the other two:
P = (2 × A) + B
A and B are the lengths of the two equal sides and the third side, respectively.
The primary formula for the triangle's perimeter can also be used to calculate this.
As seen above, any triangle's perimeter is equal to the sum of its sides. On the other hand, the two sides of an isosceles triangle have the same length, so A = C.
Therefore, P = (2 × A) + B = A + B + C = Perimeter = A + B + A.
The Right Angle Triangle's Perimeter
The easiest way to calculate a right-angled triangle's area is to add its sides. But occasionally, one or both sides might not be known. To find the perimeter, we must, therefore, use a few theorems or utilize a few formulas. Depending on the information at hand, we can apply different theorems. We need to use Pythagoras' Theorem to solve for a single unknown side. On the other hand, the law of sines must be used if two sides are unknown.
Using Pythagoras' Theorem
According to Pythagoras' theorem, the square of the hypotenuse of a right-angled triangle equals the sum of the squares of the other two sides.
C hypotenuse of the triangle.
Where the other two sides are a and b, and c is the hypotenuse.
Here's an illustration to help you comprehend.
In the figure above, a = 4, b = 3, and c are unknown. This can be found using Pythagoras' theorem.
c2 = 42+ 32.
c=√25
c=5
Isosceles Right Triangle's Perimeter
An isosceles right triangle is a triangle with one angle measuring 90° and the other two measuring 45°. As seen in the image below, it has ∠A and ∠C measuring 45° each and ∠B measuring 90°. AC and AB are equal sides. It is also called a 45-90-45 triangle.
The hypotenuse is side AC, which is the opposite of B. And one can interpret AB or AC as the base or height.
As we know, the perimeter of a triangle is equal to the sum of its sides.
Given that Base BC and Height AB are equal, the two sides of the right isosceles triangle are assigned the letter "B," and 'H' is regarded as the hypotenuse.
AB + BC + AC equals the perimeter.
= B + B + H
= 2B + H
The Scalene Triangle's Perimeter
A scalene triangle has three distinct side lengths and three distinct angle measurements. Because each side has a different length, it can only be calculated using the standard formula.
Scalene Triangle perimeter, P = A + B + C
where the side lengths of a triangle are denoted by the letters A, B, and C.
Calculation From Vertices
A triangle's vertex is the point where any two of its sides come together. Knowing the coordinates of a triangle's vertices allows us to calculate its perimeter. Think of the triangle ABC, where A = (x1,y1) and B = (x2,y2) and C≡(x3,y3) are the triangle's vertices. The line segments AB, BC, and CA comprise the side lengths. These can be located by applying the distance formula to the segment endpoints:
You are aware that in a plane with Cartesian coordinates A≡(x1,y1), the distance AB between two points
 and B≡(x2,y2)
 is determined by the subsequent formula:
AB=(x1-x2)2+ (y1-y2)2
The Pythagorean Theorem is the distance formula hidden away.
Similarly, we can determine how far apart points A and C and B and C are.
BC=(x2-x3)2+ (y2-y3)2
additionally
AC=(x1-x3)2+ (y1-y3)2
Now that we know the length of the sides, we can quickly calculate the perimeter by adding the distances together.
Therefore,Â
Perimeter = AB + BC + ACÂ
Law of Cosines
The law of Cosines is used when one side of the triangle is unknown.
When we know the measure of the opposing angle but don't know anyone's side, we can employ the Law of Cosines. Generally speaking, any triangle can be subject to this law.
c2=a2+b2−2ab cosy
The unknown side in the equation above is c, the known sides are a and b, and the angle across from the unknown side is y. Y will be 90° if c is the hypotenuse. Since cos(90) equals zero, the equation becomes a2+b2
Law of Sines
This law can be applied when two sides and at least two angles are known. This law can also be used in conjunction with other laws and formulas to determine the perimeter of any triangle. This law also links a triangle's sides and the sine of its angles.
asin ɑ= bsin β = csin γ
Here, the angles ɑ, β, and γ are represented by the sides a, b, and c, respectively.
Semi-Perimeter of Triangle
In geometry, a polygon's semi-perimeter equals half of its perimeter. A shape's semi-perimeter is half of its surrounding distance, which is known as the perimeter.
For each given polygon, the semi-perimeter can be calculated by dividing its circumference by two. The semi-perimeter has a derivation as simple as that of the perimeter, but it appears in formulas for triangles and other figures often enough to have its name. The semi-perimeter is usually represented by the letter s when it appears in a formula.
S = P/2
Practice Questions
What is the perimeter of the following triangle?
Answer:
- 12 units
- 10 units
- 8 units
- 7 units
2. The perimeter of an equilateral triangle is 45 inches. What is the side of the triangle?
- 22.5 inches
- 25 inches
- 15 inches
- 9 inches
3. If the perimeter of a scalene triangle is 32 feet. If the two sides are 12 feet and 7 feet, then what is the third side?
- 13 feet
- 10 feet
- 27 feet
- 20 feet
4. Find the perimeter of a right-angled isosceles triangle whose two sides are 4 inches each.
- 4+ 4√2 inches
- 32√2 inches
- 12√2 inches
-  8+4√2inches
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