A triangle is a closed two-dimensional shape with three straight sides and three angles. It is one of the fundamental shapes in geometry and can be found in many structures and designs, both in nature and in human-made objects. Triangles are defined by the length of their sides and the angles between them. Triangle come in many different types.
There are various ways to find the area of a triangle.
What is the Formula for the Area of a Triangle?
The area of a triangle is the amount of space inside the triangle, measured in square units.
If you know the base and height of the triangle it is easy to find the area of triangle.
It is simply multiply the base by the height, and then divide by 2
where the base b and height h are the two sides of the triangle that meet at a right angle. The base is the bottom side, and the height is the perpendicular distance from the base to the opposite vertex.
Area of Triangle Using Heron’s Formula
Heron’s formula is a mathematical formula used to calculate the area of a triangle. It was named after the ancient Greek mathematician Heron of Alexandria, who is believed to have discovered it.
The formula is particularly useful for finding the area of triangles where only the lengths of the sides are known, and the triangle cannot be easily classified into a specific type, such as right-angled or equilateral.
Heron’s formula states that the area of a triangle with sides of lengths a, b, and c is given by:
where s is the semiperimeter of the triangle, which is half the perimeter:
s = (a + b + c) / 2
Area of Triangle With 2 Sides and Included Angle (SAS)
If one angle and the length of the adjacent sides are known, the area can be calculated using trigonometry.
Specifically, the formula is:
Area of Triangle = (1/2)ab sin(C)
where a and b are the lengths of the adjacent sides, C is the angle between them, and sin(C) is the sine of the angle.
Area of a Right Angled Triangle
A right-angled triangle is a triangle where one angle is always 90° and left two acute angles add to 90°. Therefore, we can say the height of the right-angled triangle will be the length of the perpendicular side.
Area of a Right Triangle = A = ½ × Base × Height(Perpendicular distance)
Area of an Equilateral Triangle
In this type of triangle, all sides are equals.
Area of an Isosceles Triangle
An isosceles triangle has two sides and also two angles are equals.
How to Find the Area of a Triangle?
Here’s a step-by-step guide on how to find the area of a triangle using each of the three methods:
1. Area of Triangle when Base and Height are Given
Step 1: Identify the length of the base and the height of the triangle.
Step 2: Plug the values into the formula: Area of Triangle = (base x height) / 2.
Step 3: Multiply the base and height values, and then divide the result by 2 to get the area.
Step 4: Round the answer to the desired number of decimal places or leave it in exact form.
2. Area of Triangle when 3 Sides are Given
Step 1: Identify the lengths of all three sides of the triangle.
Step 2: Calculate the semiperimeter of the triangle: s = (a + b + c) / 2.
Step 3: Plug the values into Heron’s formula: Area of Triangle = √[s(s-a)(s-b)(s-c)].
Step 4: Simplify the expression inside the square root, take the square root, and round the answer to the desired number of decimal places or leave it in exact form.
3. Area of Triangle With 2 Sides and Included Angle (SAS) are Given
Step 1: Identify the length of two sides and the angle between them.
Step 2: Convert the angle to radians if necessary and calculate the sine of the angle: sin(C).
Step 3: Plug the values into the formula: Area of Triangle = (1/2)ab sin(C).
Step 4: Multiply the length of the two sides and the sine of the angle, and then divide the result by 2 to get the area.
Step 5: Round the answer to the desired number of decimal places or leave it in exact form.
Area of Triangle Examples
Example 1: What is the area of a triangle with base b =4 cm and height h = 5 cm?
Solution:
Using the formula of triangle,
Area of a Triangle, A = 1/2 × b × h
= 1/2 × 4 cm × 3 cm
= 2 cm × 3 cm
Area of a Triangle, A 6 cm2
Example 2: Find the area of an acute triangle with a base of 10 inches and a height of 6 inches.
Solution:
Area of acute triangle (A) = (½)× b × h sq.units
⇒ A = (½) × (10 in) × (6 in)
⇒ A = (½) × (60 in2)
area of an acute triangle A = 30 in2
Example 3: Find the area of a right-angled triangle with a base of 8 cm and a height of 11 cm.
Solution:
Area = (½) × b × h sq.units
⇒ Area = (½) × (8 cm) × (11 cm)
⇒ Area = (½) × (88 cm2)
area of a right-angled triangle = 44 cm2
Example 4: If, in ∆ABC, A = 30° and b = 2, c = 4 in units. Then the area will be;
Area (∆ABC) = ½ bc sin A
= ½ (2) (4) sin 30
Area = 4 x ½ (since sin 30 = ½)
= 2 sq.unit.
Example 5: If we have a triangle with sides of length 7 cm, 8 cm, and 9 cm. Find the area useing Heron’s formula.
First, we calculate the semiperimeter:
s = (7 cm + 8 cm + 9 cm) / 2
s = 12 cm
Next, we substitute the values of a, b, c, and s into the formula for the area:
Area of Triangle = √[12(12-7)(12-8)(12-9)]
Triangle = √[12(5)(4)(3)]
Area of Triangle = √[720]
≈ 26.83 cm^2