Area of Triangle
You have already learnt about the area of rectangles and squares. Now it is time to learn how to find the area of a triangle, one of the most basic shapes in geometry.
Think of cutting a rectangle along its diagonal. You get two triangles. Each triangle is exactly half the rectangle. This simple idea gives us the formula for the area of a triangle: it is half the area of the rectangle formed by its base and height.
Triangles are everywhere around us. The roof of a house, a slice of pizza, the yield sign on roads, the face of a pyramid, and even the sail of a boat are all shaped like triangles. Knowing how to calculate the area of a triangle is useful in construction, art, science, and everyday life.
In this chapter, you will learn the formula for the area of a triangle, understand what base and height mean, and practise finding the area of different types of triangles using solved examples.
What is Area of Triangle?
Definition: The area of a triangle is the amount of space enclosed within its three sides. It is measured in square units (sq. cm, sq. m, etc.).
Base and Height:
- The base of a triangle is any one of its sides (usually the bottom side).
- The height (or altitude) is the perpendicular distance from the base to the opposite vertex (the highest point).
- Every triangle has 3 possible base-height pairs. You can choose any side as the base, and the height is the perpendicular drawn from the opposite vertex to that base.
Important:
- The height must be perpendicular (at 90 degrees) to the base.
- In a right-angled triangle, two sides are already perpendicular, so one leg can be the base and the other the height.
- In an obtuse triangle, the height may fall outside the triangle when drawn from certain vertices. The formula still works the same way.
Area of Triangle Formula
Formula for Area of a Triangle:
Area of Triangle = 1/2 x base x height
Where:
- base (b) = the length of the chosen side of the triangle
- height (h) = the perpendicular distance from the base to the opposite vertex
Why this formula works:
A triangle is exactly half of a parallelogram (or rectangle) with the same base and height.
- Area of rectangle = base x height
- A diagonal divides the rectangle into 2 equal triangles
- So, area of each triangle = (1/2) x base x height
Shorthand notation:
A = 1/2 x b x h
Units:
- If base and height are in cm, area is in sq. cm (cm squared).
- If base and height are in m, area is in sq. m (m squared).
- Always make sure base and height are in the same unit before calculating.
Types and Properties
The area formula works for ALL types of triangles. Here is how to apply it for each type:
1. Right-Angled Triangle
The two sides that form the right angle can be used as base and height directly. No need to draw a separate height.
- If the two legs are 6 cm and 8 cm, Area = 1/2 x 6 x 8 = 24 sq. cm.
2. Acute-Angled Triangle
All angles are less than 90 degrees. The height drawn from any vertex falls inside the triangle.
- Choose any side as the base. Drop a perpendicular from the opposite vertex to the base. Measure the height.
3. Obtuse-Angled Triangle
One angle is more than 90 degrees. The height from the vertex of the obtuse angle falls outside the triangle (the base must be extended).
- The formula still works: Area = 1/2 x base x height.
All three sides are equal. The height can be found using the Pythagoras theorem:
Height of equilateral triangle = (root 3 / 2) x side
Area = 1/2 x side x (root 3 / 2) x side = (root 3 / 4) x side squared
Two sides are equal. The height drawn from the vertex angle (the angle between the equal sides) to the unequal side bisects the base.
Summary Table:
| Triangle Type | Base | Height | Formula |
|---|---|---|---|
| Right-angled | One leg | Other leg | 1/2 x leg1 x leg2 |
| Acute / Obtuse | Any side | Perpendicular from opposite vertex | 1/2 x base x height |
| Equilateral (side = a) | a | (root 3 / 2) x a | (root 3 / 4) x a squared |
Solved Examples
Example 1: Basic Area Calculation
Problem: Find the area of a triangle with base 12 cm and height 7 cm.
Solution:
Given:
- Base (b) = 12 cm
- Height (h) = 7 cm
Using the formula:
- Area = 1/2 x b x h
- = 1/2 x 12 x 7
- = 1/2 x 84
- = 42 sq. cm
Answer: The area of the triangle is 42 sq. cm.
Example 2: Right-Angled Triangle
Problem: A right-angled triangle has legs of length 9 cm and 12 cm. Find its area.
Solution:
Given:
- Leg 1 = 9 cm (take as base)
- Leg 2 = 12 cm (take as height, since legs are perpendicular)
Using the formula:
- Area = 1/2 x 9 x 12
- = 1/2 x 108
- = 54 sq. cm
Answer: The area of the right-angled triangle is 54 sq. cm.
Example 3: Finding the Height
Problem: The area of a triangle is 60 sq. cm and its base is 15 cm. Find the height.
Solution:
Given:
- Area = 60 sq. cm
- Base = 15 cm
Using the formula:
- Area = 1/2 x base x height
- 60 = 1/2 x 15 x height
- 60 = 7.5 x height
- height = 60 / 7.5
- height = 8 cm
Answer: The height of the triangle is 8 cm.
Example 4: Finding the Base
Problem: A triangle has area 35 sq. m and height 10 m. Find the base.
Solution:
Given:
- Area = 35 sq. m
- Height = 10 m
Using the formula:
- 35 = 1/2 x base x 10
- 35 = 5 x base
- base = 35 / 5
- base = 7 m
Answer: The base of the triangle is 7 m.
Example 5: Converting Units
Problem: Find the area of a triangle with base 1.5 m and height 80 cm.
Solution:
Given:
- Base = 1.5 m = 150 cm (converting to same unit)
- Height = 80 cm
Using the formula:
- Area = 1/2 x 150 x 80
- = 1/2 x 12000
- = 6000 sq. cm
Converting to sq. m:
- 6000 sq. cm = 6000 / 10000 = 0.6 sq. m
Answer: The area is 6000 sq. cm or 0.6 sq. m.
Example 6: Garden Plot Problem
Problem: A triangular garden has a base of 20 m and a height of 14 m. If 1 sq. m of garden costs Rs. 50 to maintain per year, what is the total annual maintenance cost?
Solution:
Step 1: Find the area.
- Area = 1/2 x 20 x 14
- = 1/2 x 280
- = 140 sq. m
Step 2: Find the cost.
- Cost = 140 x 50 = Rs. 7000
Answer: The annual maintenance cost is Rs. 7000.
Example 7: Comparing Areas of Two Triangles
Problem: Triangle A has base 16 cm and height 9 cm. Triangle B has base 12 cm and height 14 cm. Which triangle has a greater area?
Solution:
Area of Triangle A:
- = 1/2 x 16 x 9 = 72 sq. cm
Area of Triangle B:
- = 1/2 x 12 x 14 = 84 sq. cm
Comparison:
- 84 > 72
Answer: Triangle B has a greater area by 12 sq. cm.
Example 8: Triangular Sail Problem
Problem: A triangular sail on a boat has a base of 4 m and a height of 6 m. How much cloth is needed to make the sail?
Solution:
Given:
- Base = 4 m
- Height = 6 m
Using the formula:
- Area = 1/2 x 4 x 6
- = 1/2 x 24
- = 12 sq. m
Answer: 12 sq. m of cloth is needed.
Example 9: Doubling the Base
Problem: A triangle has base 10 cm and height 6 cm. If the base is doubled and the height remains the same, what happens to the area?
Solution:
Original area:
- = 1/2 x 10 x 6 = 30 sq. cm
New base = 2 x 10 = 20 cm, height = 6 cm:
- New area = 1/2 x 20 x 6 = 60 sq. cm
Comparison:
- 60 / 30 = 2
Answer: When the base is doubled, the area also doubles.
Example 10: Triangle Inside a Rectangle
Problem: A rectangle has length 18 cm and breadth 10 cm. A triangle is drawn inside the rectangle with the same base (18 cm) and the same height (10 cm). Find the area of the triangle.
Solution:
Given:
- Base of triangle = 18 cm
- Height of triangle = 10 cm
Using the formula:
- Area of triangle = 1/2 x 18 x 10
- = 1/2 x 180
- = 90 sq. cm
Note: Area of rectangle = 18 x 10 = 180 sq. cm. The triangle is exactly half the rectangle.
Answer: The area of the triangle is 90 sq. cm.
Real-World Applications
The area of a triangle is used in many real-life situations:
Construction: Builders calculate the area of triangular roofs, gable walls, and support beams to estimate the amount of material needed.
Farming: Farmers with triangular plots of land need to know the area to plan irrigation, fertiliser quantities, and crop yields.
Art and Design: Artists and designers use triangular shapes in logos, patterns, and decorations. Knowing the area helps plan layouts.
Packaging: Triangular prism-shaped boxes (like Toblerone chocolate packaging) require area calculations for the triangular faces.
Navigation: In surveying and mapping, large areas of land are divided into triangles, and the total area is found by adding up the areas of all the triangles. This method is called triangulation.
Science: The concept of area of a triangle is used in physics (force diagrams), engineering (structural analysis), and computer graphics (rendering 3D objects using triangular meshes).
Key Points to Remember
- The area of a triangle = 1/2 x base x height.
- The base can be any side. The height is the perpendicular distance from the base to the opposite vertex.
- A triangle is half of a rectangle (or parallelogram) with the same base and height.
- In a right-angled triangle, the two legs serve as base and height.
- The height must always be perpendicular to the base.
- Area is always in square units (sq. cm, sq. m, etc.).
- Make sure base and height are in the same unit before calculating.
- If the base is doubled (height constant), the area doubles.
- If both base and height are doubled, the area becomes 4 times.
- Every triangle has 3 base-height pairs, but the area is the same regardless of which pair is used.
Practice Problems
- Find the area of a triangle with base 14 cm and height 9 cm.
- A right-angled triangle has legs of 5 cm and 12 cm. Find its area.
- The area of a triangle is 48 sq. cm and its base is 16 cm. Find the height.
- The area of a triangle is 70 sq. m and its height is 14 m. Find the base.
- A triangular park has a base of 30 m and height of 20 m. Find the cost of planting grass at Rs. 8 per sq. m.
- Find the area of a triangle with base 2.4 m and height 50 cm. (Convert to the same unit first.)
- If the height of a triangle is tripled and the base remains the same, how does the area change?
- A rectangular plot is 40 m long and 25 m wide. Find the area of the triangle formed by drawing a diagonal.
Frequently Asked Questions
Q1. What is the formula for the area of a triangle?
Area of a triangle = 1/2 x base x height. The base is any side of the triangle and the height is the perpendicular distance from that base to the opposite vertex.
Q2. Why do we use 1/2 in the formula?
A triangle is exactly half of a rectangle (or parallelogram) with the same base and height. Since the area of a rectangle is base x height, the area of a triangle is half of that, which gives 1/2 x base x height.
Q3. Can any side be the base?
Yes. You can choose any of the three sides as the base. The height is then the perpendicular distance from that chosen base to the vertex opposite to it. The area will be the same no matter which side you pick as the base.
Q4. What is the height of a triangle?
The height (or altitude) of a triangle is the perpendicular distance from a vertex to the opposite side (the base). It forms a 90-degree angle with the base. In an obtuse triangle, the height from certain vertices may fall outside the triangle.
Q5. How do you find the area of a right-angled triangle?
In a right-angled triangle, the two sides forming the right angle (the legs) are already perpendicular. Use one leg as the base and the other as the height. Area = 1/2 x leg 1 x leg 2.
Q6. What happens to the area if both the base and height are doubled?
If both base and height are doubled, the new area = 1/2 x (2b) x (2h) = 1/2 x 4bh = 4 x (1/2 x b x h) = 4 times the original area.
Q7. What units are used for area?
Area is measured in square units. If base and height are in centimetres, area is in sq. cm (cm squared). If they are in metres, area is in sq. m. Always ensure base and height are in the same unit before calculating.
Q8. How is the area of an equilateral triangle found?
For an equilateral triangle with side 'a', the height is (root 3 / 2) x a. So, Area = (root 3 / 4) x a squared. For example, if each side is 6 cm, Area = (root 3 / 4) x 36 = 9 root 3 sq. cm, which is approximately 15.59 sq. cm.
