Area of Right Triangle
The area of a right triangle is the portion that is covered by the boundary of the triangle. A rightangled triangle is a triangle in which one of the angles is a right angle (90 degrees). It is simply known as a right triangle. One reallife example of a righttriangle is the triangle shape formed by a ladder on the floor that is leaned to a wall. In a rightangled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are called legs. The two legs can be interchangeably called base and height.
We have the formula to find the area of a right triangle in the above image. But what exactly the area of a right triangle means? How to derive this formula? Let us learn all these with a few solved examples and practice questions.
1.  What Is Area of a Right Triangle? 
2.  Area of Right Triangle Formula 
3.  How to Derive Area of Right Triangle Formula? 
4.  Area of Right Triangle with Hypotenuse 
5.  FAQs on Area of a Right Triangle 
What Is Area of a Right Triangle?
The area of a rightangled triangle, as we discussed earlier, is the space that is inside it. This space is divided into squares of unit length and the number of unit squares that are inside the right triangle is its area. The area is measured in square units. For example, some units of area are square inches, square meters, square kilometers, etc. Let us consider the following right triangle whose base is 4 units and whose height is 3 units.
Can you try counting the number of unit squares inside this triangle? There are 6 unit squares in total. So the area of the above triangle is 6 square units. But it is not possible to calculate the area of a right triangle always by counting the number of squares. There must be a formula to do this. Let us see what is the formula for finding the area of a right triangle.
Area of Right Triangle Formula
In the above example, if we multiply the base and height, we get 3 × 4 = 12 and if we divide it by 2, we get 6. So the area of a right triangle is obtained by multiplying its base and height and then making the product half.
Area of a right triangle = 1/2 × base × height
Examples
 The area of a right triangle with base 6 cm and height 4 cm is 1/2 × 6 × 4 = 12 cm^{2}.
 The area of a right triangle with base 10 m and height 5 m is 1/2 × 10 × 5 = 25 m^{2}.
 The area of a right triangle with base 11 in and height 5 in is 1/2 × 11 × 5 = 27.5 in^{2}.
How To Derive Area of Right Triangle Formula?
Consider a rectangle of length l and width w. Also, draw a diagonal. You can see that the rectangle is divided into two right triangles.
We know that the area of a rectangle is length × width. So the area of the above rectangle is l × w. We can see that the two right triangles are congruent as they can be arranged such that one overlaps the other. Thus, the area of the rectangle is equal to twice the area of one of the above right triangles. i.e.,
Area of rectangle = l × w = 2 × (Area of one right triangle)
This gives,
Area of one right triangle = 1/2 × l × w.
We usually represent the legs of the rightangled triangle to be base and height.
Thus, the formula for the area of a right triangle is,
Area of a right triangle = 1/2 × base × height
Area of Right Triangle With Hypotenuse
Let us recollect the Pythagoras theorem which states that in a rightangled triangle, the square of the hypotenuse is the sum of the squares of the other two sides. i.e.,
(hypotenuse)^{2} = (base)^{2} + (height)^{2}
Though it is not possible to find the area of a right triangle just with the hypotenuse, it is possible to find its area if we know one of the base and height along with the hypotenuse. Let us see an example problem.
Example: Find the area of a right triangle whose base is 6 in and whose hypotenuse is 10 in.
Solution:
Substitute the given values in the Pythagoras theorem,
(hypotenuse)^{2} = (base)^{2} + (height)^{2}
10^{2} = 6^{2} + (height)^{2}
100 = 36 + (height)^{2}
(height)^{2} = 64
height = √(64) = 8 in.
So the area of the given triangle = 1/2 × base × height = 1/2 × 6 × 8 = 24 in^{2}.
Solved Examples on Area of Right Triangle

Example 1: The longest side of a bread slice that resembles a right triangle is 13 units. If its height is 12 units, find its area using the area of a right triangle formula.
Solution:
We know that the longest side of a right triangle is called the hypotenuse.
So hypotenuse = 13 units and height = 12 units.
Substitute the given values in the Pythagoras theorem,
(hypotenuse)^{2} = (base)^{2} + (height)^{2}
13^{2} = (base)^{2} + (12)^{2}
169 = (base)^{2} + 144
(base)^{2} = 25
base = √(25) = 5 units.
The area of the bread slice = 1/2 × base x height = 1/2 × 5 x 12 = 30 square units.
Therefore, the area of the given bread slice = 30 square units.

Example 2: A swimming pool is in the shape of a right triangle. Its sides are in the ratio 3:4:5. Its perimeter is 720 m. Find its area.
Solution:
Let us assume that the sides of the swimming pool be 3x, 4x, and 5x.
It is given that its perimeter = 720 m.
3x + 4x + 5x = 720
12x = 720
x = 60
So the sides of the triangle are,
3x = 3(60) = 180 m
4x = 4(60) = 240 m
5x = 5(60) = 300 m
Since 300 m is the longest side of the swimming pool (which is in the shape of a right triangle), it is the hypotenuse.
So 180 m and 240 m must be the base and the height of the swimming pool interchangeably.
Using area of a right triangle formula,
The area of the swimming pool = 1/2 × base × height = 1/2 × 180 × 240 = 21,600 m^{2}.
Therefore, the area of the given swimming pool = 21,600 m^{2}.
FAQs on Area of Right Triangle
What Is the Area of a Right Triangle?
The area of a right triangle is defined as the total space or region covered by a rightangled triangle. It is expressed in square units. Some common units used to represent area are m^{2}, cm^{2}, in^{2}, yd^{2}, etc.
What Is the Formula for Finding the Area of a Right Triangle?
The area of a right triangle of base b and height h is 1/2 × base × height (or) 1/2 × b × h.
How Do You Find the Perimeter and Area of a Right Triangle?
The area of a right triangle of base b and height h is found using the formula 1/2 × b × h and its perimeter is obtained by just adding all the sides. In case, only two of its sides are given, then we use Pythagoras' theorem to find the third side.
How Do You Find the Area of a Right Triangle Without the Base?
If only the height and hypotenuse of a right triangle are given, then before finding the area of the triangle, we first need to find the base using the Pythagoras theorem. Then we can use the formula 1/2 × base × height to find its area.
For example, to find the area of a right triangle with a height of 4 cm and hypotenuse 5 cm, we first find its base using the Pythagoras theorem. Then we get
base = √[(hypotenuse)^{2}  (height)^{2}] = √(5^{2}  4^{2}) = √9 = 3 cm.
Area of the triangle = 1/2 × 3 × 4 = 6 cm^{2}.
How Do You Find the Area of a Right Triangle Without the Height?
If only the base and hypotenuse of a right triangle are given, then before finding the area of the triangle, we first need to find the height using the Pythagoras theorem. Then we can use the formula 1/2 × base x height to find its area.
For example, to find the area of a right triangle with a base of 4 cm and hypotenuse 5 cm, we first find its height using the Pythagoras theorem. Then we get
height = √[(hypotenuse)^{2}  (base)^{2}] = √(5^{2}  4^{2}) = √9 = 3 cm.
Area of the triangle = 1/2 × 3 × 4 = 6 cm^{2}.
How Do You Find the Area of a Right Triangle With a Hypotenuse?
In fact, it is not possible to find the area of a right triangle just with the hypotenuse. We need to know at least one of the base and height along with the hypotenuse to find the area.
 If we know the base and the hypotenuse, we find the height using the Pythagoras theorem.
 If we know the height and the hypotenuse, we find the base using the Pythagoras theorem.
Then we can find the area of the right triangle using the formula 1/2 × base × height.