In Geometry, triangles are classified based on sides and angles. The right triangle, also known as the right-angled triangle, is one of the types of a triangle that is classified based on angle, where one of its angles is equal to 90 degrees (Right angle). In this article, we will learn the definition of the right triangle, the area of right triangle, formulas and examples in detail.
Table of Contents:
- Right Triangle Definition
- Area of Right Triangle
- Calculating the Hypotenuse of Right Triangle
- Examples
- Practice Questions
- FAQs
What is the Right Triangle?
As discussed above, the right triangle is a triangle in which one of its angles is equal to 90°. In the right triangle, the side opposite to the right angle is called the hypotenuse, whereas the other two sides are called the legs of the right-angle triangle. The legs are interchangeably called the base (adjacent side) and the height (perpendicular side).
What is the Area of Right Triangle?
The area of a right triangle is the space occupied inside the boundary of the right triangle. Generally, the space inside the boundary is divided into squares of unit length. Hence, the number of unit squares that are present inside the right triangle is calculated as the area of a right triangle. The unit used to measure the area is square units.
Area of Right Triangle Formula
The formula to calculate the area of a right triangle is given by:
Area of Right Triangle, A = (½) × b × h square units
Where,
“b” is the base (adjacent side)
“h” is the height (perpendicular side)
Hence, the area of the right triangle is the product of base and height and then divide the product by 2.
Derivation for Area of Right Triangle
To derive the formula for the area of a right triangle, let us consider a rectangle of length “l” and width “w”. Now, draw a diagonal as shown in the below figure.
From the figure, it is observed that a rectangle is divided into two right-angled triangles, and they are congruent to each other, such that one triangle overlaps the other triangle.
We know that,
Area of a rectangle = Length × Width square units
So, the area of rectangle = 2 × (Area of one right triangle)
Thus, the Area of one right triangle = (½) × Area of rectangle = (½ ) × length × Width
Since, length = base (b) and width = height(h),
The area of a right triangle = (½)×b×h square units
How to Calculate the Hypotenuse of a Right Triangle?
The hypotenuse of the right triangle can be calculated using the Pythagoras theorem. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
(i.e) (Hypotenuse)2 = (Base)2 + (Height)2.
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Area of Right Triangle Examples
Example 1:
The longest side of a right triangle is 17 cm and the height is 15 cm. Find the area of the right triangle.
Solution:
Given:
The longest side of a right triangle is 17 cm = Hypotenuse
Height = 15 cm.
To find the area of a right triangle, first, we need to find the base of the right triangle.
Finding the Base of a Right Triangle:
Using Pythagoras theorem, the base can be calculated as follows:
(Hypotenuse)2 = (Base)2 + (Height)2
(17)2 = (Base)2 + (15)2
(Base)2 = 172 – 152
(Base)2 =289 – 225
(Base)2 = 64
Hence, Base = √64 = 8 cm.
Therefore, the base of the right triangle is 8 cm.
Finding the Area of a Right Triangle:
Area of right triangle = (½)×b×h square units
Substituting the values in the formula, we get
A = (½)×8×15 cm2
A = 4×15 cm2
A = 60 cm2
Therefore, the area of the right triangle is 60 cm2.
Example 2:
Calculate the height of the right triangle, whose base length is 60 m and area is 420 m2.
Solution:
Given:
Base = 60 m
Area = 420 m2
The formula for the area of a right-angle triangle is A = (½)×b×h square units.
Now, substitute the values in the formula
420 = (½)×60×h
420 = 30×h
h = 420/30
h = 14 m
Therefore, the height of the right triangle is 14 m.
Practice Questions
Solve the following problems:
- A field is in the shape of a right triangle and its sides are in the ratio of 3:4:5. Find the area of the field, given that the perimeter is 720 units.
- Find the area of the right triangle whose base is 10 inches and height is 5 inches.
- What is the base of the right triangle whose height is 4 m and the area is 12 m2?
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Frequently Asked Questions on Area of Right Triangle
What is the area of right triangle?
The area of a right triangle is the region occupied inside the boundary of the right-angled triangle.
What is the formula for the area of a right triangle?
The formula to calculate the area of a right triangle is:
Area of right triangle = (½) × Base × Height square units
How to calculate the hypotenuse of a right triangle?
The hypotenuse of a right triangle can be calculated using the Pythagoras theorem.
i.e. (Hypotenuse)2 = (Base)2+(Height)2.
How to find the perimeter of a right triangle?
The perimeter of a right triangle is found by adding all the sides of a right triangle.
What is the area of a right triangle, whose base is 11 cm and height is 5 cm?
Given: Base = 11 cm and height = 5 cm
Area of a right triangle = (½)×b×h = (½)×11×5 = 27.5 cm2.
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