What Is the Area of a Triangle? Formulas, Examples & How to Find Missing Sides

Q: What is the formula of area of triangle?

A = ½ × base × height, with the height perpendicular to the base.

Q: How do you find the area of a triangle with three sides?

Use Heron’s formula: find s = (a + b + c)/2, then A = √[s(s − a)(s − b)(s − c)].

Q: How do you find the area of an equilateral triangle?

Use A = (√3/4) a², where a is the side length.

Q: How do you find the area of a right-angled triangle?

Multiply the two legs and divide by 2.

Q: What is the vector formula?

A = ½ |a × b| for two side vectors from the same vertex.

Q: Can you find the sides if only the area is known?

Not for a general triangle, since many triangles share one area. An equilateral triangle is the exception.

Q: How do you find the area without the height?

Use Heron’s formula for three sides, or ½ ab sin C for two sides and the included angle.

The area of triangle is the amount of flat space enclosed by its three sides. The basic triangle area formula is A = ½ × base × height. If you want to know how to calculate the area of a triangle, start there, then pick a different formula when the question gives other information.

Triangle area matters in geometry and everyday life: roofs, sails, garden beds and land plots are triangles or can be cut into them. This guide covers every common method, step-by-step examples, how to find the side of a triangle from its area, and a short practice set.

☰ Table of Contents

What is the Area of Triangle? Meaning and Definition

The area of a triangle is the measure of the region enclosed by its three sides. It equals half the base multiplied by the perpendicular height, and it is written in square units such as cm² or m².

Base (b): the side you measure from.
Height (h): the perpendicular distance from the base to the opposite vertex. A slanted side is not the height unless it is perpendicular to the base.

Area counts unit squares inside the shape, so a length in cm becomes cm × cm = cm².

Example: base 10 cm, height 6 cm gives ½ × 10 × 6 = 30 cm².

Area of Triangle Formula – Basic Formula Explained

A = ½ × b × h, where A is the area, b is the base and h is the perpendicular height.

Picture triangle ABC with BC as the base. Drop a dotted line from A that meets BC at D with a right-angle mark. AD is the height. In an obtuse triangle D falls on the extension of the base, but AD is still perpendicular. Any side can be the base, as long as you use its matching height.

How to Find Area of Triangle Step by Step

Identify the base.
Find the perpendicular height.
Multiply base by height.
Divide by 2.
Write the answer in square units.

Example 1: b = 8 cm, h = 5 cm: 40 ÷ 2 = 20 cm².

Example 2: b = 12 m, h = 7.5 m: 90 ÷ 2 = 45 m².

Example 3: b = 15 cm, h = 9 cm: 135 ÷ 2 = 67.5 cm².

Formula of Area of Triangle for Different Types

Area of a Right-Angled Triangle

A = ½ × leg₁ × leg₂. The legs are perpendicular, so one is the base and the other the height. Legs 6 cm and 8 cm give 24 cm².

Area of an Equilateral Triangle

A = (√3/4) a². Side 10 cm gives (√3/4) × 100 = 25√3 ≈ 43.30 cm².

Area of an Isosceles Triangle

Find the height h = √(a² − b²/4), where a is each equal side and b the base, then use ½ b h. Equal sides 13 cm, base 10 cm: h = √(169 − 25) = 12, so A = ½ × 10 × 12 = 60 cm².

Area of a Scalene Triangle

No single shortcut exists. Use ½ b h if a height is given, Heron’s formula for three sides, or ½ ab sin C for two sides and the included angle. Sides 13, 14, 15 cm give 84 cm² (see Heron’s formula below).

Triangle	Formula	Use when
Any	½ b h	Base and height known
Right-angled	½ × leg₁ × leg₂	Both legs known
Equilateral	(√3/4) a²	Side known
Isosceles	½ b √(a² − b²/4)	Equal side and base known
Scalene	Heron’s formula	Three sides known

Area of Triangle with 3 Sides – Heron's Formula

When all three sides are known but the height is not, use Heron’s formula:

A = √[s(s − a)(s − b)(s − c)], where s = (a + b + c)/2

Here a, b, c are the side lengths and s is the semi-perimeter, half the perimeter. The sides must satisfy the triangle inequality: any two sides must add to more than the third. Sides 2, 3, 7 fail because 2 + 3 < 7.

Example 1: sides 5, 12, 13 cm. s = 15. A = √(15 × 10 × 3 × 2) = √900 = 30 cm².

Example 2: sides 7, 8, 9 cm. s = 12. A = √(12 × 5 × 4 × 3) = √720 = 12√5 ≈ 26.83 cm².

Example 3 (harder): sides 25, 29, 36 m. s = 45. A = √(45 × 20 × 16 × 9) = √129600 = 360 m².

Area of Triangle Equilateral – Formula and Examples

A = (√3/4) a². All three sides equal a. Derivation: the altitude splits the triangle into two right triangles with hypotenuse a and base a/2, so h = √(a² − a²/4) = (√3/2) a. Then A = ½ × a × (√3/2) a = (√3/4) a².

Side 6 cm: 9√3 ≈ 15.59 cm².
Side 2.5 cm: 1.5625√3 ≈ 2.71 cm².

The exact value is 9√3; 15.59 is a rounded approximation.

Area of Triangle in Coordinate Geometry

For vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃):

A = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|

Plot the three points on a coordinate plane and join them. The absolute value is needed because listing the points clockwise gives a negative sum, and area cannot be negative. If the area is 0, the points are collinear.

Example 1: A(1, 1), B(5, 1), C(3, 5). Sum = 1(−4) + 5(4) + 3(0) = 16, so A = 8 square units.

Example 2: A(2, 3), B(−1, 4), C(4, −2). Sum = 2(6) + (−1)(−5) + 4(−1) = 13, so A = 6.5 square units.

Area of Triangle in Vector Form

If a and b are side vectors from the same vertex, A = ½ |a × b|. The cross product’s magnitude is |a||b| sin θ, the area of the parallelogram on a and b, and a triangle is half of it.

2D: a = (4, 0), b = (1, 3). a × b = 4×3 − 0×1 = 12, so A = 6.
3D: a = (1, 2, 3), b = (2, 1, 0). a × b = (−3, 6, −3), magnitude √54 = 3√6, so A = 3√6/2 ≈ 3.67.

How to Find the Side of a Triangle Using Its Area

Important: area alone does not fix the sides of a general triangle. Base 6 with height 4 and base 8 with height 3 both give area 12. You need extra information.

Finding the Base When Area and Height Are Known

Base = 2 × Area / Height. Area 36 cm², height 9 cm: base = 72/9 = 8 cm. Area 52.5 m², height 7 m: base = 105/7 = 15 m.

Finding the Height When Area and Base Are Known

Height = 2 × Area / Base. Area 60 cm², base 15 cm: height = 8 cm. Area 45 cm², base 12 cm: height = 7.5 cm.

Finding the Side of an Equilateral Triangle from Its Area

a = √(4A/√3). Area 16√3 cm²: a² = 64, so a = 8 cm. Here the area is enough because the shape has one free measurement.

Finding a Missing Side Using the Pythagorean Theorem

In a right-angled triangle, hypotenuse² = leg² + leg². Hypotenuse 17, one leg 8: other leg = √(289 − 64) = √225 = 15.

Area of Triangle Using Two Sides and an Included Angle

A = ½ ab sin C, where C is the angle between sides a and b. It helps when the height is not given.

a = 8, b = 10, C = 30°: ½ × 80 × 0.5 = 20 cm².
a = 7, b = 9, C = 60°: ½ × 63 × (√3/2) = 63√3/4 ≈ 27.28 cm².

Area of Triangle – All Important Formulas at a Glance

Given	Formula	When to use
Base, height	½ b h	Perpendicular height known
Right-angled	½ leg₁ leg₂	Both legs known
Equilateral	(√3/4) a²	Side known
Three sides	√[s(s−a)(s−b)(s−c)]	Height unknown
Two sides, included angle	½ ab sin C	Angle between the sides
Coordinates	½ \|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)\|	Points on a plane
Vectors	½ \|a × b\|	Two side vectors
Missing base	2A / h	Area, height known
Missing height	2A / b	Area, base known
Equilateral side	√(4A/√3)	Area known

Solved Examples on Area of Triangle

Isosceles: equal sides 10 cm, base 12 cm. h = √(100 − 36) = 8, so A = ½ × 12 × 8 = 48 cm².
Heron’s: sides 11, 13, 20 cm. s = 22, A = √(22 × 11 × 9 × 2) = √4356 = 66 cm².
Coordinates: A(−2, 1), B(4, 3), C(1, −4). Sum = −14 − 20 − 2 = −36, so A = ½ × 36 = 18 square units.
Missing base: area 84 cm², height 12 cm. Base = 168/12 = 14 cm.
Word problem: a triangular garden has base 18 m and height 10 m, and seed costs ₹40 per m². Area = 90 m², cost = ₹3,600.
Word problem: a triangular gable wall is 8 m wide and 3 m high. Area = 12 m²; at 4 m² per litre, one coat needs 3 litres.

Common Mistakes While Calculating the Area of a Triangle

Forgetting ½: b = 10, h = 6 gives 30, not 60.
Using the slanted side: base 10, slant 13, height 12 gives ½ × 10 × 12 = 60, not 65.
Mixed units: base 2 m and height 50 cm: convert to 0.5 m, so A = 0.5 m².
Missing square units: write 24 cm², not 24 cm.
Wrong s in Heron’s formula: for 3, 4, 5 use s = 6, not the perimeter 12.
Dropping the absolute value: a sum of −36 gives area 18.
Assuming area fixes the sides: area 12 fits many triangles.

Easy Tricks to Remember Triangle Area Formulas

Two identical triangles form a parallelogram, so a triangle is half of base × height.
Always draw the perpendicular height first.
Match the formula to the given information using the table above.
Estimate to check: base 10, height 6 sits inside a 10 × 6 rectangle, so the area must be under 60.

Area of Triangle Practice Questions with Answers

Easy

Base 10 cm, height 7 cm. Find the area.
Area 48 cm², height 8 cm. Find the base.
Right triangle with legs 5 cm and 12 cm. Find the area.
Equilateral triangle, side 12 cm. Find the area.

Intermediate

Sides 8, 15, 17 cm. Find the area.
Isosceles, equal sides 17 cm, base 16 cm. Find the area.
Right triangle, hypotenuse 25 cm, one leg 7 cm. Find the other leg and the area.
Area 64 cm², base twice the height. Find both.

Advanced

Vertices (0, 0), (8, 0), (3, 6). Find the area.
Show (1, 2), (4, 6), (7, 10) are collinear.
Side vectors (5, 2) and (1, 4). Find the area.
Sides 12 cm and 9 cm with included angle 30°. Find the area.

Answers: (1) 35 cm². (2) 12 cm. (3) 30 cm². (4) 36√3 ≈ 62.35 cm². (5) s = 20, √3600 = 60 cm². (6) h = 15, area 120 cm². (7) Leg 24 cm, area 84 cm². (8) h² = 64, so height 8 cm, base 16 cm. (9) 24 square units. (10) The sum is −4 + 32 − 28 = 0, so the area is 0 and the points are collinear. (11) ½(5×4 − 2×1) = 9 square units. (12) ½ × 12 × 9 × 0.5 = 27 cm².

Conclusion

The area of a triangle is the space inside its sides, measured in square units. Remember A = ½ × base × height first, then choose Heron’s formula, the trigonometric formula, or the coordinate and vector methods depending on what you are given. Work through the practice questions and check each answer.

Frequently Asked Questions

What is the formula of area of triangle?