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This lesson will show you different ways to find the area of triangles. You could start by taking the quiz below to see how much you know. By definition, the area of a triangle is the amount of space taken by the triangle in a two-dimensional plane.
The most common way to find the area of a triangle is to use the basic formula shown below.
Area of a triangle = (base × height) ÷ 2
The red line represents the length of the base (call it b if you want) and the blue line represents the length of the height.
As you can see, to find the area, just multiply the base of the triangle by the height of the triangle and divide the result by 2.
Draw a rectangle around the triangle with the same base and height. The two yellow pieces outside the triangle fit together exactly to make a copy of the triangle. So the triangle is half of the rectangle:
rectangle = b × h, so triangle = ½ × b × h
The height always meets the base at a right angle. In an acute triangle it is inside the triangle. In a right triangle it is one of the two sides that form the right angle. In an obtuse triangle it falls outside: extend the base with a dashed line and measure down to it.
However, you can also find the area of a triangle based on the following two cases:
The area is expressed in square units.
Find the area of the triangle shown in the figure below with base 9 inches and height 4 inches:
Area = (base × height) ÷ 2
Area = (9 × 4) ÷ 2
Area = 36 ÷ 2
Area = 18
The area is 18 square inches or 18 in².
Find the area of the obtuse triangle shown in the figure below with base 4 inches and height 3 inches:
Area = (base × height) ÷ 2
Area = (4 × 3) ÷ 2
Area = 12 ÷ 2
Area = 6
The area is 6 square inches or 6 in².
Find the area of the right-angled triangle shown in the figure below with base 5 inches and height 6 inches:
Area = (base × height) ÷ 2
Area = (5 × 6) ÷ 2
Area = 30 ÷ 2
Area = 15
The area is 15 square inches or 15 in².
If you know the area and one of the two lengths, turn the formula around:
h = (2 × A) ÷ b and b = (2 × A) ÷ h
Try it: a triangle has an area of 24 cm² and a base of 8 cm. Find its height.
h = (2 × 24) ÷ 8
h = 48 ÷ 8 = 6
The height is 6 cm.
The ancient Greek mathematician Heron is most famous for this formula for the area of a triangle when you know the lengths of its sides a, b, and c.
The area is the square root of the product s(s − a)(s − b)(s − c).
A = √[s(s − a)(s − b)(s − c)]
where s is the semi-perimeter of the triangle:
s = (a + b + c) ÷ 2
The lengths of the sides of a triangle are 15 feet, 9 feet, and 12 feet. Use Heron's formula to find the area of the triangle.
s = (a + b + c) ÷ 2
s = (15 + 9 + 12) ÷ 2
s = 36 ÷ 2
s = 18
A = √[18(18 − 15)(18 − 9)(18 − 12)]
A = √[18(3)(9)(6)]
A = √[(54)(54)]
A = √2916
A = 54
The area is 54 square feet. Read the lesson about Heron's formula to see more examples.
When two sides and the angle between the two sides are known (SAS), the area is half the product of the adjacent sides times the sine of the included angle.
Therefore, use the formula below to find the area:
Area = [bc(sin A)] ÷ 2
The lengths of two adjacent sides of a triangle are 8 cm and 12 cm. The angle between the sides is 60 degrees. Find the area of the triangle.
Let b = 8 cm, c = 12 cm, and A = 60°
Area = [bc(sin A)] ÷ 2
Area = [8 × 12(sin 60°)] ÷ 2
Area = [96(sin 60°)] ÷ 2
Area = [96(0.866)] ÷ 2
Area = 83.136 ÷ 2
Area ≈ 41.57
The area is about 41.57 cm². Read the lesson about area of a triangle given SAS to see how the formula was derived and 1 more example showing how to use the formula.
In an equilateral triangle all three sides have the same length s, so A = (√3 ÷ 4) × s². Find the area of an equilateral triangle with sides of 6 cm.
A = (√3 ÷ 4) × 6²
A = (√3 ÷ 4) × 36 = 9√3
A ≈ 15.59
The area is about 15.59 cm². See the lesson about the area of an equilateral triangle for more.
| You know… | Use |
|---|---|
| the base and the height | A = ½ × b × h |
| all three sides (SSS) | Heron's formula |
| two sides and the angle between them (SAS) | A = ½ × b × c × sin A |
| the side of an equilateral triangle | A = (√3 ÷ 4) × s² |