The
Area of Isosceles triangle
formula represents the space it occupies within a two-dimensional plane. The standard formula for calculating the area of a triangle is half the product of its base and height. In this discussion, we provide a comprehensive explanation of the area of an isosceles triangle, including its formula and derivation, and include solved examples to enhance your comprehension of this concept.
What is the Area of Isosceles Triangle
Formula
?
The area of an isosceles triangle is the cumulative space enclosed by the shape. In the case of an isosceles triangle, calculating its area is straightforward when you have both the height (also referred to as the altitude) and the length of the base. To determine the area of the isosceles triangle, simply multiply the height by the base and divide the result by 2.
What is an Isosceles Triangle?
An isosceles triangle is characterized by the presence of two sides that have equal lengths, which also corresponds to having two angles of the triangle equal in measure. In essence, an isosceles triangle possesses two identical sides and two congruent angles. The term "isosceles" originates from the Greek words "iso" (meaning same) and "skelos" (meaning leg). It's worth noting that an equilateral triangle is a special case of the isosceles triangle, where all three sides and angles of the triangle are equal.
When it comes to determining the area of an isosceles triangle, this geometric shape's defining features include two sides of equal length and two angles that are congruent. The vertex where these equal sides meet the third side creates a symmetrical structure. By drawing a perpendicular line from the intersection point of the two equal sides to the base of the unequal side, we can generate two right-angle triangles.
Formula for Calculating the Area of an Isosceles Triangle
To determine the area of an isosceles triangle, you can utilize the following formula:
Area = ½ × base × Height
Additionally,
The perimeter of the isosceles triangle can be calculated as follows:
Perimeter P = 2a + b
The altitude of the isosceles triangle can be determined using the formula:
Altitude h = √(a
2
− b
2
/4)
These formulas provide essential tools for computing the area, perimeter, and altitude of an isosceles triangle based on its specific characteristics.
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List of Formulas to Find Isosceles Triangle Area
Using the base and height:
Area (A) = ½ × base (b) × height (h)
where 'b' represents the base length, and 'h' is the height.
Using all three sides:
A = ½[√(a
2
− b
2
⁄4) × b]
where 'a' denotes the measure of the equal sides, and 'b' is the base of the triangle.
Using the length of two sides and an angle between them:
A = ½ × a × b × sin(α)
where 'a' represents the length of the equal sides, 'b' is the base of the triangle, and 'α' is the angle between them.
Using two angles and the length between them:
A = [a
2
×sin(β)×sin(α)/ 2×sin(2π−α−β)]
where 'a' represents the length of the equal sides, 'α' is the measure of equal angles, and 'β' is the angle opposite to the base.
Area formula for an isosceles right triangle:
A = ½ × a
2
where 'a' is the measure of the equal sides.
These formulas offer various approaches to calculate the area of an isosceles triangle based on the available information about its sides, angles, and base.
How to Calculate Area if Only Sides of an Isosceles Triangle are Known?
When you have information about the length of the equal sides and the length of the base of an isosceles triangle, you can calculate the height or altitude of the triangle using this formula:
Altitude of an Isosceles Triangle =
√(a
2
− b
2
/4)
Consequently, you can determine the area of the isosceles triangle solely based on its sides with this formula:
Area of Isosceles Triangle Using Only Sides =
½[√(a
2
− b
2
/4) × b]
Here's a breakdown of the variables:
'b' represents the base of the isosceles triangle.
'h' denotes the height or altitude of the isosceles triangle.
'a' indicates the length of the two equal sides.
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Derivation for Isosceles Triangle Area Using Heron’s Formula
Deriving the Area of an Isosceles Triangle Using Heron's Formula can be explained as follows:
Heron's Formula states:
Area = √[s(s−a)(s−b)(s−c)]
Where, 's' is the semi-perimeter, given by s = ½(a + b + c).
Now, for an isosceles triangle:
s = ½(a + a + b)
⇒ s = ½(2a + b)
Or, s = a + (b/2)
Hence,
Area = √[s(s−a)(s−b)(s−c)]
Or, Area = √[s (s−a)² (s−b)]
⇒ Area = (s−a) × √[s (s−b)]
By substituting the value of 's':
⇒ Area = (a + b/2 − a) × √[(a + b/2) × ((a + b/2) − b)]
⇒ Area = b/2 × √[(a + b/2) × (a − b/2)]
Or, the area of an isosceles triangle can be expressed as:
Area = b/2 × √(a² − b²/4)
This derivation demonstrates how to find the area of an isosceles triangle using Heron's Formula, given the lengths of its sides.
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Solved Examples
Example 1: Calculate the area of an isosceles triangle when the base (b) is 12 cm and the height (h) is 17 cm.
Solution:
Base of the triangle (b) = 12 cm
Height of the triangle (h) = 17 cm
Area of the Isosceles Triangle = (1/2) × b × h
= (1/2) × 12 × 17
= 6 × 17
= 102 cm²
Example 2: Determine the length of the base of an isosceles triangle with an area of 243 cm² and an altitude (h) of 27 cm.
Solution:
Area of the triangle (A) = 243 cm²
Height of the triangle (h) = 27 cm
Base of the triangle (b) = ?
Area of the Isosceles Triangle = (1/2) × b × h
243 = (1/2) × b × 27
243 = (b × 27) / 2
b = (243 × 2) / 27
b = 18 cm
Hence, the base of the triangle is 18 cm.
Example 3: Find the area, altitude, and perimeter of an isosceles triangle with a = 5 cm (length of two equal sides) and b = 9 cm (base).
Solution:
Given:
a = 5 cm
b = 9 cm
Perimeter of the isosceles triangle:
P = 2a + b
P = 2(5) + 9 cm
P = 10 + 9 cm
P = 19 cm
Altitude of the isosceles triangle:
h = √(a² − b²/4)
h = √(5² − 9²/4)
h = √(25 − 81/4) cm
h = √(25 − 20.25) cm
h = √4.75 cm
h = 2.179 cm
Area of the isosceles triangle:
A = (b × h)/2
A = (9 × 2.179)/2 cm²
A = 19.611/2 cm²
A ≈ 9.806 cm²
Example 4: Calculate the area, altitude, and perimeter of an isosceles triangle with a = 12 cm and b = 7 cm.
Solution:
Given:
a = 12 cm
b = 7 cm
Perimeter of the isosceles triangle:
P = 2a + b
P = 2(12) + 7 cm
P = 24 + 7 cm
P = 31 cm
Altitude of the isosceles triangle:
h = √(a² − b²/4)
h = √(12² − 7²/4)
h = √(144 − 49/4) cm
h = √(144 − 12.25) cm
h = √131.75 cm
h ≈ 11.478 cm
Area of the isosceles triangle:
A = (b × h)/2
A = (7 × 11.478)/2 cm²
A = 80.346/2 cm²
A ≈ 40.173 cm²
