The area of an isosceles triangle is the amount of region enclosed by it in a two-dimensional space. The general formula for the area of triangle is equal to half the product of the base and height of the triangle. Here, a detailed explanation of the isosceles triangle area, its formula and derivation are given along with a few solved example questions to make it easier to have a deeper understanding of this concept. Show
Check more mathematics formulas here. What is the Formula for Area of Isosceles Triangle?The total area covered by an isosceles triangle is known as its area. For an isosceles triangle, the area can be easily calculated if the height (i.e. the altitude) and the base are known. Multiplying the height with the base and dividing it by 2, results in the area of the isosceles triangle. What is an isosceles triangle? An isosceles triangle is one that has at least two sides of equal length. This property is equivalent to two angles of the triangle being equal. An isosceles triangle has two equal sides and two equal angles. The name derives from the Greek iso (same) and Skelos (leg). An equilateral triangle is a special case of the isosceles triangle, where all the three sides and angles of the triangle are equal.  An isosceles triangle has two equal side lengths and two equal angles, the corners at which these sides meet the third side is symmetrical in shape. If a perpendicular line is drawn from the point of intersection of two equal sides to the base of the unequal side, then two right-angle triangles are generated. Table of Contents: Area of Isosceles Triangle FormulaThe area of an isosceles triangle is given by the following formula: Also,
List of Formulas to Find Isosceles Triangle Area
How to Calculate Area if Only Sides of an Isosceles Triangle are Known?If the length of the equal sides and the length of the base of an isosceles triangle are known, then the height or altitude of the triangle is to be calculated using the following formula:
Thus,
Here,
Derivation for Isosceles Triangle Area Using Heron’s FormulaThe area of an isosceles triangle can be easily derived using Heron’s formula as explained below. According to Heron’s formula, Area = √[s(s−a)(s−b)(s−c)] Where, s = ½(a + b + c) Now, for an isosceles triangle, s = ½(a + a + b) ⇒ s = ½(2a + b) Or, s = a + (b/2) Now, Area = √[s(s−a)(s−b)(s−c)] Or, Area = √[s (s−a)2 (s−b)] ⇒ Area = (s−a) × √[s (s−b)] 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, area of isosceles triangle = b/2 × √(a2 − b2/4) Area of Isosceles Right Triangle Formula
Derivation: Area = ½ ×base × height area = ½ × a × a = a2/2 Perimeter of Isosceles Right Triangle FormulaDerivation: The perimeter of an isosceles right triangle is the sum of all the sides of an isosceles right triangle. Suppose the two equal sides are a. Using Pythagoras theorem the unequal side is found to be a√2. Hence, perimeter of isosceles right triangle = a+a+a√2 = 2a+a√2 = a(2+√2) = a(2+√2) Area of Isosceles Triangle Using TrigonometryUsing Length of 2 Sides and Angle Between Them A = ½ × b × c × sin(α) Using 2 Angles and Length Between Them A = [c2×sin(β)×sin(α)/ 2×sin(2π−α−β)] Solved ExamplesExample 1: Find the area of an isosceles triangle given b = 12 cm and h = 17 cm? Base of the triangle (b) = 12 cm Height of the triangle (h) = 17 cm Area of Isosceles Triangle = (1/2) × b × h = (1/2) × 12 × 17 = 6 × 17 = 102 cm2 Example 2: Find the length of the base of an isosceles triangle whose area is 243 cm2, and the altitude of the triangle is 27 cm. Solution: Area of the triangle = A = 243 cm2 Height of the triangle (h) = 27 cm The base of the triangle = b =? Area of Isosceles Triangle = (1/2) × b × h 243 = (1/2) × b × 27 243 = (b×27)/2 b = (243×2)/27 b = 18 cm Thus, the base of the triangle is 18 cm. Question 3: Find the area, altitude and perimeter of an isosceles triangle given a = 5 cm (length of two equal sides), b = 9 cm (base). Solution: Given, a = 5 cm b = 9 cm Perimeter of an isosceles triangle = 2a + b = 2(5) + 9 cm = 10 + 9 cm = 19 cm Altitude of an isosceles triangle h = √(a2 − b2/4) = √(52 − 92/4) = √(25 − 81/4) cm = √(25–81/4) cm = √(25−20.25) cm = √4.75 cm h = 2.179 cm Area of an isosceles triangle = (b×h)/2 = (9×2.179)/2 cm² = 19.611/2 cm² A = 9.81 cm² Question 4: Find the area, altitude and perimeter of an isosceles triangle given a = 12 cm, b = 7 cm. Solution: Given, a = 12 cm b = 7 cm Perimeter of an isosceles triangle = 2a + b = 2(12) + 7 cm = 24 + 7 cm P = 31 cm Altitude of an isosceles triangle = √(a2 − b2⁄4) = √(122−72/4) cm = √(144−49/4) cm = √(144−12.25) cm = √131.75 cm h = 11.478 cm Area of an isosceles triangle = (b×h)/2 = (7×11.478)/2 cm² = 80.346/2 cm² = 40.173 cm² Practice Questions
An isosceles triangle can be defined as a special type of triangle whose 2 sides are equal in measure. For an isosceles triangle, along with two sides, two angles are also equal in measure.
The area of an isosceles triangle is defined as the amount of space occupied by the isosceles triangle in the two-dimensional plane.
To calculate the area of an isosceles triangle, the following formula is used: A = ½ × b × h
The formula to calculate the perimeter of an isosceles triangle is: P = 2a + b
The area of an isosceles triangle is the amount of space enclosed between the sides of the triangle. Besides the general area of the isosceles triangle formula, which is equal to half the product of the base and height of the triangle, different formulas are used to calculate the area of triangles, depending upon their classification based on sides. These different types based on sides are given below:
Let us understand the area of the isosceles triangle in detail in the following section. What is Area of an Isosceles Triangle?The area of an isosceles triangle is the total space or region covered between the sides of an isosceles triangle in two-dimensional space. An isosceles triangle is defined as a triangle having two sides equal, which also means two equal angles. Here are some properties of an Isosceles triangle that distinguish it from other types of triangles:
The area of an isosceles triangle is expressed in square units. Therefore, some units that can be used to represent the area of an isosceles triangle are m2, cm2, in2, yd2, etc. Area of an Isosceles Triangle FormulasThe area of an isosceles triangle refers to the total space covered by the shape in 2-D. The area of an isosceles triangle can be calculated in many ways based on the known elements of the isosceles triangle. The general basic formula that can be used to calculate the area of an isosceles triangle using height is given as, (1/2) × Base × Height The following table summarizes different formulas that can be used to calculate the area of an isosceles triangle, for a different set of known parameters.
where,
Area of Isosceles Triangle Using SidesIf the length of the equal sides and the base of an isosceles triangle are known, then the height or altitude of the triangle can be calculated. The formula to calculate the area of an Isosceles triangle using sides is given as, Area of isosceles triangle using only sides = ½[√(a2 - b2/4) × b] where,
Derivation: From the above figure, we know: BD = DC = ½ BC = ½ b (perpendicular from the vertex angle bisects the base) Applying Pythagoras' theorem for ΔABD, we get: The altitude of an isosceles triangle = √(a2 − b2/4) Also, we know the general area of the triangle formula is given as: Substituting value for height: Area of isosceles triangle using only sides = ½[√(a2 − b2 /4) × b] Isosceles Triangle Area Using Heron’s FormulaThe area of an isosceles triangle formula can be easily derived using Heron’s formula as explained in the following steps. Heron's formula is used to find the area of a triangle when the measurements of its 3 sides are given. Derivation: The Heron's formula to find the area, A of a triangle whose sides are a,b, and c is: A = √s(s-a)(s-b)(s-c) where,
We know that the perimeter of a triangle with sides a, b, and c is a + b + c. Here, s is half of the perimeter of the triangle, and hence, it is called semi-perimeter. Thus, the semi-perimeter is: s = (a + b + c)/2 Now, for an isosceles triangle, s = ½(a + a + b) ⇒ s = ½(2a + b) or, s = a + (b/2) Also, Area = √[s(s−a)(s−b)(s−c)] or, Area = √[s (s−a)2 (s−b)] ⇒ Area = (s−a) × √[s (s−b)] Substituting the value for “s” ⇒ Area = (a + b/2 − a) × √[(a + b/2) × ((a + b/2) − b)] ⇒ Area = b/2 × √[(a + b/2) × (a − b/2)] Area of isosceles triangle = b/2 × √(a2 − b2/4) square units where,
Isosceles Triangle Area Using Trigonometry(SAS and ASA)The formula to find area of an isosceles triangle using length of 2 sides and angle between them or using 2 angles and length between them can be calculated using basic trigonometry concepts. Using 2 sides and angle between them: Area = ½ × b × a × sin(α) square units where,
Using 2 angles and length between them: Area = [a2 × sin(β/2) × sin(α)] square units where,
Area of an Isosceles Right TriangleA right isosceles triangle is defined as the isosceles triangle which has one angle equal to 90°. The Formula to calculate the area for an isosceles right triangle can be expressed as, Area = ½ × a2 where a is the length of equal sides. Derivation: Let the equal sides of the right isosceles triangle be denoted as "a", as shown in the figure below: The length of the hypotenuse, BC can be calculated using Pythagoras' Theorem, BC2 = a2 + a2 BC = √2 a Area = ½ × base × height Area = ½ × a × a = a2/2 square units
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FAQs on Area of an Isosceles TriangleThe area of a figure is the region enclosed by the figure. Thus, the area of an isosceles triangle means the total space enclosed by an isosceles triangle. What is an Isosceles Triangle?An isosceles triangle is defined as a triangle having two sides equal, which also means two equal angles. Here are some properties of an Isosceles triangle that distinguish it from other types of triangles:
What is the Formula for Area of Isosceles Triangle?The area of an isosceles triangle refers to the total space covered by the shape in 2-D. The area of an isosceles triangle can be calculated in many ways based on the known elements of the isosceles triangle.
where, b = base of the isosceles triangle a = measure of equal sides of the isosceles triangle α = measure of equal angles of the isosceles triangle β = measure of the angle opposite to the base How Do You Find the Height Using Area of an Isosceles Triangle?The height or altitude of an Isosceles Triangle can be calculated by applying the Pythagorean Theorem for any two sides. The formula to calculate the height of an isosceles triangle is given as, where, b = base of the isosceles triangle a = measure of equal sides of the isosceles triangle What is the Perimeter and Area of an Isosceles Triangle?The perimeter of an isosceles triangle is defined as the length of the boundary of an isosceles triangle. The formula for the perimeter of an isosceles triangle is given as, P = 2a + b units. While area is the total region covered by the isosceles triangle, given as, ½[√(a2 − b2 ⁄4) × b]
How Do You Find the Area of an Isosceles Triangle Without Height?The expression to calculate the area of an isosceles triangle without height can be calculated using Heron's formula. The formula to calculate the area of an isosceles triangle without height is given as,
How Do You Find the Area of an Isosceles Triangle Given Two Sides and an Angle?The area of a triangle is half the product of the given two sides and sine of the included angle. Area of Triangle with 2 Sides and Included Angle (SAS) formula is used to find the general formula for calculating the area of an isosceles triangle for SAS as, Area = ½ × b × a × sin(α) where,
How Do You Find the Area of an Isosceles Triangle With 3 Sides?The area of an isosceles triangle with 3 sides can be calculated using Heron's formula, that is Area = \(\sqrt {s(s - a)(s - b)(s - c)} \). For an isosceles triangle, side c = side a. The general formula to calculate the area of an isosceles triangle is given as,
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An isosceles triangle with perimeter is 40 and the two sides are 17 and 8 cm then its area is sq cm
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