Area Of A Equal Triangle
Area of Equilateral Triangle
The area of an equilateral triangle is the corporeality of space that an equilateral triangle covers in a 2-dimensional airplane. An equilateral triangle is a triangle with all sides equal and all its angles measuring 60º. The area of any shape is the number of unit squares that tin can fit into it. Hither, "unit" refers to one (1) and a unit of measurement square is a square with a side of i unit. Alternatively, the surface area of an equilateral triangle is the total amount of space it encloses in a 2-dimensional plane.
| 1. | What is the Area of an Equilateral Triangle? |
| two. | Area of Equilateral Triangle Formula |
| iii. | Derivation of Surface area of Equilateral Triangle |
| 4. | How to Detect Surface area of Equilateral Triangle? |
| 5. | FAQs on Area of Equilateral Triangle |
What is the Area of an Equilateral Triangle?
The expanse of an equilateral triangle is divers as the region covered within the three sides of the triangle. Information technology is expressed in square units. Some important units used to express the area of an equilateral triangle are in2, k2, cmii, ydii, etc. Let united states of america understand the formula to calculate the area of an equilateral triangle and its derivation in the sections below.
Expanse of an Equilateral Triangle Formula
The equilateral triangle area formula is used to calculate the infinite occupied between the sides of the equilateral triangle in a 2d plane. Computing areas of any geometrical shape is a very important skill used past many people in their work. For the equilateral triangle, nosotros have the formula for its area. In a general triangle, finding the area of a triangle might be a little bit complicated for certain reasons. But, finding the area of an equilateral triangle is quite an like shooting fish in a barrel calculation.
The general formula for the area of a triangle whose base and meridian are known is given as:
Area = 1/2 × base × acme
While the formula to calculate the surface area of an equilateral triangle is given equally,
Area = √3/four × (side)ii foursquare units
In the given triangle ABC, Area of ΔABC = (√3/four) × (side)2 square units, where, AB = BC = CA = a units (the length of equal sides of the triangle).
Thus, the formula for the area of the above equilateral triangle can be written as:
Area of equilateral triangle ΔABC = (√3/4) × a2 square units
Case: How to observe the area of an equilateral triangle with one side of 4 units?
Solution:
Using the area of equilateral triangle formula: (√3/four) × a2 square units,
nosotros volition substitute the values of the side length.
Therefore, the surface area of the equilateral triangle (√3/iv) × four2 = iv√iii square units.
Derivation of Area of an Equilateral Triangle Formula
In an equilateral triangle, all the sides are equal and all the internal angles are sixty°. So, an equilateral triangle's expanse tin can be calculated if the length of one side is known. The formula to summate the expanse of an equilateral triangle is given as,
Area of an equilateral triangle = (√3/4) × aii square units
where,
a = Length of each side of an equilateral triangle
The above formula to find the surface area of an equilateral triangle can be derived in the following means:
- Using the general expanse of a triangle formula
- Using Heron's formula
- Using trigonometry
Deriving Equilateral Triangle'southward Area Using Expanse of Triangle Formula
The formula used to calculate the expanse of an equilateral triangle can be derived using the general area of the triangle formula. To do so, we require the length of each side and the acme of the equilateral triangle. We will thus calculate the summit of an equilateral triangle in terms of the side length.
The formula for the area of an equilateral triangle comes out from the general formula of the area of the triangle which is equal to ½ × base × height. Derivation for the formula of an equilateral triangle is given beneath.
Surface area of triangle = ½ × base × peak
For finding the pinnacle of an equilateral triangle we employ the Pythagoras theorem (hypotenuse2 = base2 + heighttwo).
Here, base of operations = a/2, height = h, and hypotenuse = a (refer to the to a higher place effigy).
At present, apply the Pythagoras theorem in the triangle.
aii = h2 + (a/ii)2
⇒ h2 = a2 – (a2/4)
⇒ h2 = (3a2)/iv
Or, h = ½(√3a)
At present, put the value of "h" in the area of the triangle equation.
Area of Triangle = ½ × base × height
⇒ A = ½ × a × ½(√3a) [The base of operations of the triangle is 'a' units]
Or, area of equilateral triangle = ¼(√3aii)
So, the formula of height comes as ½ × (√iii × side), and farther, the area of the equilateral triangle becomes √3/4 × side2 square units.
Deriving Area of Equilateral Triangle Using Heron's Formula
Heron'southward formula is used to find the area of a triangle when the lengths of the iii sides of the triangle are known. In mathematics, Heron's formula is named after Hero of Alexandria, who gives the expanse of any triangle when the lengths of all three sides are known. We do not utilise angles or other distances for finding the area of a triangle using Heron'southward formula.
Steps are given below for finding the area of a triangle:
Consider the triangle ABC with sides a, b, and c. Heron's formula to find the surface area of the triangle is:
Area = √s(s - a)(s - b)(s - c)
where,
s is the semi-perimeter which is given by:
s = (a + b + c)/2
For equilateral triangle: a = b = c.
south = (a + a + a)/ii
due south = 3a/ii
Now, Surface area of equilateral triangle = \(\sqrt {south(s - a)(southward - a)(s - a)}\)
= \(\sqrt {\frac{{3a}}{2}(\frac{{3a}}{two} - a)(\frac{{3a}}{2} - a)(\frac{{3a}}{2} - a)}\)
= \(\sqrt {\frac{3a}{two}(\frac{{a}}{2})(\frac{{a}}{2})(\frac{{a}}{2})}\)
= \(\sqrt{\frac{3a}{2}(\frac{{a^iii}}{eight}})\)
=\(\sqrt{({\frac{{3a^four}}{sixteen}})}\)
Area of equilateral triangle = √3/4 × (side)2 square units.
Deriving Area of Equilateral Triangle With 2 Sides and Included Angle (SAS)
For finding the area of a triangle with 2 sides and the included angle, use the sine trigonometric part to calculate the tiptop of a triangle and use that value to discover the area of the triangle. At that place are three variations to the aforementioned formula based on which sides and included angle are given.
Consider a, b, and c are the different sides of a triangle.
- When sides 'b' and 'c' and included bending A is known, the area of the triangle is: one/2 × bc × sin(A)
- When sides 'b' and 'a' and included angle C is known, the surface area of the triangle is: 1/ii × ab × sin(C)
- When sides 'a' and 'c' and included angle B is known, the area of the triangle is: i/2 × ac × sin(B)
In an equilateral triangle, ∠A = ∠B = ∠C = 60°. Therefore, sin A = sin B = sin C. Now, area of △ABC = ane/2 × b × c × sin(A) = 1/2 × a × b × sin(C) = 1/2 × a × c × sin(B).
For equilateral triangle, a = b = c (refer to the above effigy).
Area = 1/2 × a × a × sin(C) = one/ii × a2 × sin(threescore°) = 1/2 × aii × √3/2
So, area of equilateral triangle = (√3/iv)a2 square units.
How to Find the Area of Equilateral Triangle?
The following steps tin can be followed to discover the surface area of an equilateral triangle using the side length:
- Step ane: Note the mensurate of the side length of the equilateral triangle.
- Step 2: Apply the formula to calculate the equilateral triangle's surface area given every bit, A = (√iii/four)a2, where, a is the mensurate of the side length of the equilateral triangle.
- Step three: Express the answer with the advisable unit.
Now, that we accept learned the formula and method to calculate the area of the equilateral triangle, let united states see a few solved examples for meliorate agreement.
Examples on Area of Equilateral Triangle
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Practise Questions on Area of Equilateral Triangle
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FAQs on Expanse of an Equilateral Triangle
What is the Surface area of an Equilateral Triangle in Math?
The expanse of an equilateral triangle in math is the region encompassed or enclosed within the three sides of the equilateral triangle. It is expressed in square units or (unit)2.
What is the Formula of Equilateral Triangle Area?
We can calculate the area of an equilateral triangle given the length of each side. The formula of equilateral triangle area is equal to √3/4 times of square of the side length of the equilateral triangle.
☛ As well Check:
- Triangle Formula
- Equilateral Triangle Formulas
- Area Formulas
How do you Discover the Perimeter and Area of an Equilateral Triangle?
The area of an equilateral triangle is √3/4 × (side)2 square units and the perimeter of an equilateral triangle is three times a side of the equilateral triangle.
How practice you Calculate the Elevation Using Surface area of an Equilateral Triangle?
Given the area of an equilateral triangle, we tin can notice the mensurate of each side using the formula, Area = √3/4 × (side)2. For finding the meridian of an equilateral triangle from the side length, we utilise the Pythagoras theorem (hypotenusetwo = base2 + heighttwo). So, the formula of height comes as ½ × (√3 × side).
How to Find the Sides of an Equilateral Triangle if the Area of an Equilateral Triangle is known?
If the area of an equilateral triangle is known, nosotros put the given value in the following formula and solve for the length of the side:
Expanse of equilateral triangle = (√3/4)a2 where a is the length of the side of the equilateral triangle.
What is the Use of the Area of Equilateral Triangle Calculator?
The area of an equilateral triangle reckoner is an online tool used to determine the expanse. This is the quickest mode to calculate the area of an equilateral triangle by providing an input value such as the length of the side. Try Cuemath's expanse of an equilateral triangle calculator now and calculate the area in a few seconds.
What is the Area of Equilateral Triangle with Side 2 cm?
The surface area of an equilateral triangle with a side 2 cm is given as Expanse = √3/4 × (side)2. By substituting the value of side as 2, we become Area = √3/4 × (2)2 = √iii cmtwo. Therefore, the area of an equilateral triangle with a side length of 2 cm is √iii square centimeters.
How to Discover Area of Equilateral Triangle without Tiptop?
When the pinnacle of the triangle is non given, and then the equilateral triangle surface area can exist calculated using its side length by using the formula Area = √3/4 × (side)2 square units.
How to Observe Area of Equilateral Triangle with Perimeter?
When the perimeter of an equilateral triangle is known, then we tin first detect the side of the triangle by dividing the perimeter by 3. So, we can find the area of the equilateral triangle by using the formula √3/4 × (side)2.
Area Of A Equal Triangle,
Source: https://www.cuemath.com/measurement/area-of-equilateral-triangle/
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