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December 25, 201539 Insightful Publications

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**“, “

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.

**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.

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°.

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.

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 cm ^{2}= 150 cm ^{2} |

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 unit^{2}. What is the area of triangle ABC?

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

**Q1: What are Right Angled Triangles?**

**Q2: What is the formula for finding the area of a right angle triangle?**

**Q3: How do you find the area and perimeter of a right triangle?**

**Q4: What is the sum of all the interior angles of right triangle?**

**Q5: How Do You Find the Area of a Right Triangle With a Hypotenuse?**

**Related Concepts**:

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