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  • Last Modified 17-01-2023

Area of Right Angled Triangle: Definition, Formula, Examples

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A Right-Angled Triangle is one of the most important shapes in geometry and is the basis of trigonometry. A right-angled triangle is a triangle that has three sides, namely “base“, “hypotenuse“, and “height“, with the angle between base and height being 90°. The fraction of a right triangle that is covered inside the triangle’s edge is its area. A right-angled triangle is one in which one of the angles is the same as the other (90 degrees). It’s just called a right triangle. The hypotenuse is the side opposite the right angle of a right-angled triangle, while the other two sides are termed legs. The terms “base” and “height” can be used interchangeably to describe the two legs. It is important for students to know Area of Right Angled Triangle There are mainly two formulas to calculate the area of a right-angle triangle. We will look into both these formulas in detail. As Mathematics is a tough subject, students need to spend more time learning and revising the topics. With regular practice, speed can be developed. This will they will be able to attempt more questions in a shorter time span and score higher marks.

Maths consists of Algebra and Geometry which hold equal importance. It is essential for students to learn the basic concepts in high school, as the concepts are interlinked with each other in higher classes. Continue reading to know about Area of Right Angled Triangle- Definition, Formula, Examples and more.

Area of Right Angle Triangle: Definition & Formulas

Definition of Right Triangle: A right triangle is a regular polygon, with three sides and three angles, one of the angles measuring 90°. This is a unique property of a right-angled triangle. As with all other types of triangles, the sum of all the three internal angles equals to 180°.

The area of a right angle triangle is the total area enclosed in between the sides of the triangle. There are two formulas to calculate the area depending on whether we have all the sides given or just base and height. Let’s see the two formulas in detail.

Area of Right Angle Triangle: Using Base & Height

We can calculate the area of a right-angled triangle when we have the base and the height.

Area of Right Angle Triangle = ½ x Base x Perpendicular

Derivation:

Start with a rectangle ABCD and let h be the height and b be the base as shown below:

The area of this rectangle is b × h

However, if we draw a diagonal from one vertex, it will break the rectangle into two congruent or equal right triangles.

Since the diagonal of a rectangle bisects it into two equal and congruent triangles, the area of each triangle is half the area of the rectangle. Hence,

Area (△ABC) = Area(△ACD) = ½ bh

The area of a right triangle when only angles are given:

For a right-angled triangle, the base is always perpendicular to the height. When the sides of the triangle are not given, and only angles are given, the area of a right-angled triangle can be calculated by the given formula:

$$Area = {{bc \times ba} \over 2}$$

Where a, b, c are respective angles of the right-angle triangle, with ∠b always being 90°.

Area of Right Angle Triangle Using Heron’s Formula

When all the sides of a right-angle triangle are given, we can use Heron’s Formula to calculate the area. The formula for the area of a right triangle is given by:

$$ Area = \sqrt {s(s – a)(s – b)(s – c)}$$

Where, s is the semi perimeter and is calculated as $$ s = {{(a + b + c)} \over 2}$$ and a, b, c are the sides of a triangle.

Solved Examples on Area of Right Angle Triangle

Here we have provided some of the questions on the area of the right triangle along with solutions.

Q1: Find the area of right-angled triangle whose hypotenuse is 15 cm and one of the sides is 12 cm.
A: AB² = AC² – BC² 
= 15² – 12² 
= 225 – 144
= 81
Therefore, AB = 9
Hence, area of the triangle = ¹/₂ × base × height
= ¹/₂ × 12 × 9 
= 54 cm²
Q2: The base and height of the triangle are in the ratio 3:2. If the area of the triangle is 243 cm² find the base and height of the triangle.
A: Let the common ratio be x.
Then height of triangle = 2x 
And the base of triangle = 3x
Area of triangle = 243 cm²
= 1/2 × b × h
⇒ 243 = 1/2 × 3x × 2x
⇒ 3x² = 243
⇒ x² = 243/3
⇒ x = √81
⇒ x = √(9 × 9) 
⇒ x = 9
Therefore, height of triangle = 2 × 9 
= 18 cm 
Base of triangle = 3x 
= 3 × 9 
= 27 cm
Q3: Find the area of a triangle whose sides are 41 cm, 28 cm, 15 cm. Also, find the length of the altitude corresponding to the largest side of the triangle.
A: Semi-perimeter of the triangle = (a + b + c)/2
= (41 + 28 + 15)/2 
= 84/2 
= 42 cm
Therefore, area of the triangle = √(s(s – a) (s – b) (s – c)) 
= √(42 (42 – 41) (42 – 28) (42 – 15)) cm²
= √(42 × 1 × 27 × 14) cm²
= √(3 × 3 × 3 × 3 × 2 × 2 × 7 × 7) cm²
= 3 × 3 × 2 × 7 cm²
= 126 cm²
Now, area of triangle = 1/2 × b × h 
Therefore, h = 2A/b
= (2 × 126)/41
= 252/41
= 6.1 cm
Q4: The sides of the triangular plot are in the ratio 2 : 3 : 4 and the perimeter is 180 m. Find its area.
A: Let the common ratio be x, then the three sides of triangle are 2x, 3x, 4x
Now, perimeter = 180 m
Therefore, 2x + 3x + 4x = 180
⇒ 9x = 180
⇒ x = 180/9
⇒ x = 20
Therefore, 2x = 2 × 20 = 40
3x = 3 × 20 = 60
4x = 4 × 20 = 80
Area of triangle = √(s(s – a) (s – b) (s – c))
= √(90(90 – 80) (90 – 60) (90 – 40))
= √(90 × 10 × 30 × 50))
= √(3 × 3 × 2 × 5 × 2 × 5 × 3 × 2 × 5 × 5 × 5 × 2)
= 3 × 2 × 5 × 2 × 5 √(3 × 5)
= 300 √15 m²
= 300 × 3.872 m²
= 1161.600 m²
= 1161.6 m²
Q5: What is the area of a right angle triangle whose base = 15cm and height = 20cm.
A: Area of triangle = 1/2 x base x height
Putting the values of base and height in the above equation, we get:
Area = 1/2 x 15 x 20 cm2
= 150 cm2

Practice Questions on Area of Right Angle Triangle

Here are some important questions on the area of the right triangle for you to practice:

Question 1: Find the area and perimeter of a right-angled triangle with sides of lengths 0.4 ft and 0.3 ft.
Question 2: Find the area and perimeter of an isosceles right-angled triangle with a hypotenuse of length 50 cm.
Question 3: The first side of a right-angled triangle is 200 m longer than the second side. Its hypotenuse has a length of 1000 m. Find the lengths of the two sides, the area and the perimeter of this triangle.
Question 4: A right-angled triangle has one side that is 4/3 of the second. Its hypotenuse has a length of 30 ft. Find the sides, the area and the perimeter of this triangle.
Question 5: ABC is a right triangle with a perimeter equal to 60 units and an area is equal to 150 units 2. Find its two sides and hypotenuse.
Question 6: The right-angled triangles shown below have angles ACB and DFE equal in size. The ratio of the hypotenuse AC and DF are AC/DF = 3/2. The area of triangle DEF is 100 unit2. What is the area of triangle ABC?

FAQs

Here are some of the frequently asked questions on the area of the right triangle:

Q1: What are Right Angled Triangles?
Ans: Right-angled triangles are those triangles in which one angle is 90 degrees. Since one angle is 90°, the sum of the other two angles will be 90°.

Q2: What is the formula for finding the area of a right angle triangle?
Ans: The formula to find the area of a right triangle is 1/2 x base x perpendicular.

Q3: How do you find the area and perimeter of a right triangle?
Ans: The perimeter of a right triangle is calculated by adding the measures of all sides of the triangle. Suppose a, b, and c are the sides of a right-angle triangle, then its perimeter is given as (a + b + c). The area is calculated using Heron’s formula or base and height.

Q4: What is the sum of all the interior angles of right triangle?
Ans: The total of all interior angles of any triangle equals 180 degrees.

Q5: How Do You Find the Area of a Right Triangle With a Hypotenuse?
Ans: The area of a right triangle cannot be calculated just using the hypotenuse. To find the area, we need to know at least one of the base and height, including the hypotenuse.

Related Concepts:

Area of Triangle Concepts
Area of Triangle in Coordinate Geometry
Area of Triangle & Properties
Properties of Triangle Concepts
Area of Triangle Vectors Cross Product Concepts
Important Points of Triangle
Examples of Properties of Triangle

We hope this detailed article on the area of the right triangle formula helps you. If you have any queries regarding this article, reach out to us through the comment section below and we will get back to you as soon as possible.

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