A right triangle is a triangle in which exactly one angle measures 90 degrees. Since the sum of the measures of angles in a triangle has to be 180 degrees, it is evident that the sum of the remaining two angles would be another 90 degrees. The two perpendicular sides are called the legs of a right triangle, and the longest side that lies opposite the 90-degree is called the hypotenuse of a right triangle. A right triangle can be scalene (having all three sides of different length) or isosceles (having exactly two sides of equal length). It can never be an equilateral triangle. In this article, you are going to study the definition, area, and perimeter of an isosceles right triangle in detail. Show
An Isosceles Right Triangle is a right triangle that consists of two equal length legs. Since the two legs of the right triangle are equal in length, the corresponding angles would also be congruent. Thus, in an isosceles right triangle, two legs and the two acute angles are congruent.  Since it is a right triangle, the angle between the two legs would be 90 degrees, and the legs would obviously be perpendicular to each other. Isosceles Right Triangle FormulaThe most important formula associated with any right triangle is the Pythagorean theorem. According to this theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides of the right triangle. Now, in an isosceles right triangle, the other two sides are congruent. Therefore, they are of the same length “l”. Thus, the hypotenuse measures h, then the Pythagorean theorem for isosceles right triangle would be: (Hypotenuse)2 = (Side)2 + (Side)2 h2 = l2 + l2 h2 = 2l2 Also, two congruent angles in isosceles right triangle measure 45 degrees each, and the isosceles right triangle is: Area of an Isosceles Right TriangleAs we know that the area of a triangle (A) is ½ bh square units Where b is the base of the triangle h is the altitude of the triangle In an isosceles right triangle, two legs are of equal length. Let us say that they both measure “l” then the area formula can be further modified to: Area, A = ½ (l × l) A = ½ l2 Area of an Isosceles Right Triangle = l2/2 square units. Where l is the length of the congruent sides of the isosceles right triangle Perimeter of an Isosceles Right TriangleThe perimeter of any plane figure is defined as the sum of the lengths of the sides of the figure. For a triangle, the perimeter would be the sum of all the sides of the triangle. Thus, the perimeter a triangle with side lengths a, b, and c, would be: Perimeter of a triangle = a + b + c units In an isosceles right triangle, we know that two sides are congruent. Suppose their lengths are equal to l, and the hypotenuse measures h units. Thus the perimeter of an isosceles right triangle would be: Perimeter = h + l + l units Therefore, the perimeter of an isosceles right triangle P is h + 2l units. Where h is the length of the hypotenuse side l is the length of the adjacent and opposite sides. Video Lesson on Types of TrianglesIsosceles Right Triangle ExampleQuestion: Find the area and perimeter of an isosceles right triangle whose hypotenuse side is 10 cm. Solution: Given: Length of the hypotenuse side, h = 10 cm We know that, h2 = 2l2 Substitute the value of “h” in the above formula: 102 = 2l2 100= 2l2 l2 = 100/2 l2 = 50 Therefore, l = √50 = 5√2 cm Therefore, the length of the congruent legs is 5√2 cm. So, the area of an isosceles right triangle, A = l2/2 A = (5√2)2/2 A = (25 x 2)/2 A = 25 Therefore, the area of an isosceles right triangle = 25 cm2 The perimeter of an isosceles right triangle, p = h+ 2l units P = 10 + 2( 5√2) P = 10 + 10√2 Substituting √2 = 1.414, P = 10 + 10(1.414) P = 10 + 14.14 P = 24.14 Therefore, the perimeter of an isosceles right triangle is 24.14 cm. Register with BYJU’S – The Learning App and also download the app to read all Maths-related topics and explore videos to learn with ease. 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. 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 a triangle that has any of its two sides equal in 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 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:
The 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: Altitude of an Isosceles Triangle = √(a2 − b2/4) Thus, Area of Isosceles Triangle Using Only Sides = ½[√(a2 − b2 /4) × b] 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 FormulaThe formula for Isosceles Right Triangle Area= ½ × a2 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
Frequently Asked Questions on Area of Isosceles Triangle
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 How do you find the altitude of an isosceles right triangle?Altitude of an isosceles triangle = h=√a2−b24 h = a 2 − b 2 4 ; where 'a' is one of the equal sides, 'b' is the third side of the triangle.
What is the formula for finding the altitude of a triangle?The altitude of a triangle formula for a right triangle is given as h= √xy, where x and y are the length of segments of hypotenuse divided by altitude. The altitude of the right triangle is equal to the geometric mean of the segments made by that altitude on the hypotenuse.
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What is the altitude of an isosceles right triangle with leg x
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