Area of the Triangle Formula
We can use the following formula to calculate the area of a triangle:
- Area of triangle A = \(\frac{1}{2}\) × b × h
We will discuss this formula in detail in the next section.
How do we use the area of a triangle formula?
To calculate the area of a triangle, you need the measure of its base and height.
As we just saw, the area (A) of a triangle is one-half of the product of its base (b) and height (h).
Mathematically, the area of a triangle can be expressed as:
Area of triangle A = \(\frac{1}{2}\) × b× h
Where,
b = base of a triangle
h = height of a triangle.
The area is measured in square units.
[Note : The base and the height are perpendicular to each other.]
Solved Examples
Example 1: Find the area of the triangle whose base is 7 inches long and height is 14 inches.
Solution:
\(A=\frac{1}{2}\times b \times h\) Area of a triangle formula
\(~~~=\frac{1}{2}\times 7 \times (14)\) Substitute values of b and h
\(~~~=\frac{1}{2}\times (98)\) Multiply
\( ~~~= 49 \) Simplify
Hence, the area of the triangle is 49 square inches.
Example 2: The area of the triangle is 94 sq.mm. Find the height of the triangle if the base is 12 mm long.
Solution:
The area of the triangle is 94 sq.mm and the base is 12 mm. Therefore, we use the area of a triangle formula to form an equation to find height,
\(\frac{1}{2}\times b \times h = A\) Area of a triangle formula
\(\frac{1}{2}\times (12) \times h = 94\) Substitute values of b and A
\( (12) \times h = 188 \) Multiply both sides by 2
\(h=\frac{188}{12}\) Divide both sides by 12
\(h\approx\) 15.67
Hence, the height of the triangle is approximately 15.67 mm.
Example 3: The area of the triangle is 25 sq.ft . Find the base of the triangle if the height is 5 ft long.
Solution:
The area of the triangle is 25 sq.ft and the height is 5 sq.ft. Therefore, we use the area of a triangle formula to form an equation to find the base,
\(\frac{1}{2}\times b \times h = A\) Area of a triangle formula
\(\frac{1}{2}\times b \times (5) = 25\) Substitute values of h and A
\(b \times (5) = 50\) Multiply both sides by 2
\(b=\frac{50}{5}\) Divide both sides by 5
\( b = 10 \)
Hence, the base of the triangle is 10 ft .
Example 4: Find the measure of the unknown side.
Solution:
The unknown side is the height, h, of the triangle.
The area of the triangle is 90 sq.ft and the base is 15 ft. Therefore, we use the area of a triangle formula to form an equation to find the height,
\(\frac{1}{2}\times b \times h = A\) Area of a triangle formula
\(\frac{1}{2}\times 15 \times h = 90 \) Substitute values of b and A
\((15) \times h = 180 \) Multiply both sides by 2
\(h=\frac{180}{15}\) Divide both sides by 15
\(h = 12 \)
Hence, the height of the triangle is 12 ft.
Example 5: Larry has bought a piece of land in a village and it is in the shape of a triangle as shown in the diagram. How many square yards of land did Larry buy?
Solution:
We will use the area of triangle formula to find the the number of square yards of land bought by Larry,
\(A=\frac{1}{2}\times b \times h\) Area of a triangle formula
\(~~~=\frac{1}{2} \times 35 \times (30)\) Substitute values of b and h
\(~~~=\frac{1}{2}\times 1050 \) Multiply
\(~~~= 525 \) Simplify
Hence, Larry bought 525 square yards of land.
The sum of the measures of all three sides of a triangle is the perimeter of the triangle.
A triangle in which all three sides are of different lengths is a scalene triangle.
A figure formed by combining triangles, rectangles and any other type of polygon is called a composite figure.
Example:
Any side of a triangle can be taken as its base when calculating its area.