Area of Triangle: Formulas, Examples and Problems
Area of Triangle is calculated using the base and corresponding perpendicular height. Different formulas apply when the sides, included angle, or semiperimeter are given. This page covers the standard formula, Heron's formula, right and equilateral triangle cases, comparison rules, shortcuts and solved area of triangle questions for quantitative aptitude.
On this page
What Is the Area of a Triangle?
If the base is b and the perpendicular height to that base is h, then A = 1/2 bh. The height must meet the line containing the selected base at 90°. For a base of 12 cm and height of 5 cm, A = 1/2 × 12 × 5 = 30 cm².
Area of Triangle Video Lessons
Watch short topic-wise lessons for quick revision.
Triangle Basic Formulas
Learn the basic formulas for triangles, including perimeter, area, and related measurements used in two-dimensional mensuration problems.
Area of Triangle Formula & Tricks
Important Formulas
Use b as the selected base and h as the perpendicular distance from the opposite vertex to that base.
Use this formula when all three side lengths a, b and c are known.
Use sides a and b with their included angle C. The angle must lie between the two given sides.
Here a is the common side length of an equilateral triangle.
The two perpendicular sides act as the base and height.
Quick Tricks
Any side can be selected as the base, but its corresponding perpendicular height must be used. Select the pair that gives direct or simple values.
Triangles with the same altitude have areas in the ratio of their bases: A₁/A₂ = b₁/b₂.
For side lengths a, b and c, first verify the triangle inequality: a + b > c, b + c > a and c + a > b. Then calculate the semiperimeter.
Area of Triangle Concepts
Area Using Base and Perpendicular Height
The height is not necessarily a side of the triangle. It is the shortest perpendicular distance from the opposite vertex to the selected base or its extension. If b = 15 m and h = 8 m, then A = 1/2 × 15 × 8 = 60 m².
Heron's Formula for Three Given Sides
First find the semiperimeter s = (a + b + c)/2. Then substitute in A = √[s(s − a)(s − b)(s − c)]. For sides 13 cm, 14 cm and 15 cm, s = 21 cm and A = √(21 × 8 × 7 × 6) = 84 cm².
Area from Two Sides and the Included Angle
The angle C must be between sides a and b. For sides 10 cm and 12 cm with included angle 30°, A = 1/2 × 10 × 12 × sin 30° = 30 cm².
Special Triangles
For a right triangle with legs 9 cm and 12 cm, A = 1/2 × 9 × 12 = 54 cm². For an equilateral triangle with side 6 cm, A = (√3/4) × 36 = 9√3 cm².
Area Ratios and Changes in Dimensions
If both base and height are multiplied by k, the area is multiplied by k². If only the base is doubled, the area doubles. If both dimensions are doubled, the area becomes four times the original.
Area of Triangle Visual Guide & Data
Use the table and chart below to understand the data better.
Common Triangle Area Formulas
Select the formula according to the information given in the question.
| Given information | Formula | Condition or note |
|---|---|---|
| Base and perpendicular height | A = 1/2 bh | Height must correspond to the selected base |
| Three sides | A = √[s(s − a)(s − b)(s − c)] | s = (a + b + c)/2 |
| Two sides and included angle | A = 1/2 ab sin C | C lies between sides a and b |
| Three equal sides | A = (√3/4)a² | Equilateral triangle |
| Two perpendicular sides | A = 1/2 xy | Right triangle |
Note: Area is expressed in square units, such as cm², m² or km².
Triangle with Base and Perpendicular Height
Learn more concepts on PrepShots
Start ₹1 Trial →Practice Area of Triangle Questions
Practise published questions related to this topic.
Want more practice questions?
Practice in App →Area of Triangle Quick Quiz
Attempt 5 questions and check your score instantly.
Keep practising
Practice more Area of Triangle questions in the PrepShots app and continue from your current topic.
Practice More Questions - Start ₹1 Trial →Quick Revision Notes
Area of Triangle Revision Points
Recall the formula, the required measurements and the correct units before solving.
- A = 1/2 × base × corresponding perpendicular height.
- Heron's formula is A = √[s(s − a)(s − b)(s − c)], with s = (a + b + c)/2.
- For two sides and included angle, A = 1/2 ab sin C.
- A right triangle uses its two perpendicular sides in A = 1/2 xy.
- An equilateral triangle of side a has area (√3/4)a².
- Area is measured in square units.
- If both base and height are multiplied by k, area is multiplied by k².
- For equal heights, areas are proportional to bases; for equal bases, areas are proportional to heights.
Area of Triangle FAQs
What is the area of a triangle with base 14 cm and height 9 cm?
A = 1/2 × 14 × 9 = 63 cm².
Which height is used in the triangle area formula?
Use the perpendicular height corresponding to the selected base. It must meet the base or its extension at 90°.
How is the area of a triangle found when all three sides are given?
Use Heron's formula: A = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2.
What is the area of a triangle with sides 3 cm, 4 cm and 5 cm?
It is a right triangle, so A = 1/2 × 3 × 4 = 6 cm².
What is the area formula when two sides and the included angle are given?
Use A = 1/2 ab sin C, where C is the angle between sides a and b.
If the base and height of a triangle are both doubled, how does its area change?
The area becomes four times the original because A is proportional to b × h, giving a factor of 2 × 2 = 4.