Area of Triangle (Half Base Height)
You already know how to find the area of a rectangle (length × width). Now, what about a triangle? A triangle is one of the most common shapes in mathematics and in the world around you — the face of a pyramid, the sail of a boat, and the roof of a house are all triangles.
The area of a triangle is exactly half the area of the rectangle that encloses it. If you draw a rectangle and cut it diagonally, each half is a triangle. This gives us the simple formula: Area = 1/2 × base × height.
This formula works for all triangles — whether they are right-angled, isosceles, equilateral, or scalene.
What is Area of Triangle?
Definition: The area of a triangle is the amount of space enclosed within its three sides.
Formula:
Area of Triangle = 1/2 × base × height
Where:
- Base (b): Any side of the triangle.
- Height (h): The perpendicular distance from the base to the opposite vertex (top point).
- The height must be perpendicular (at 90°) to the base.
Why half?
A rectangle with the same base and height has area = base × height. A triangle is exactly half of such a rectangle. So area of triangle = 1/2 × base × height.
Types and Properties
1. Right-angled Triangle
The two shorter sides (legs) can serve as base and height, since they are already perpendicular.
- Base = one leg, Height = other leg.
- Area = 1/2 × leg₁ × leg₂.
2. Acute Triangle
The height falls inside the triangle. Drop a perpendicular from the top vertex to the base.
The height may fall outside the triangle. You extend the base line and drop the perpendicular to the extended line.
4. Any Side Can Be the Base
You can choose any side of the triangle as the base. The height will change accordingly, but the area remains the same.
Solved Examples
Example 1: Example 1: Basic triangle area
Problem: Find the area of a triangle with base 8 cm and height 5 cm.
Solution:
- Area = 1/2 × base × height
- = 1/2 × 8 × 5
- = 1/2 × 40 = 20 cm²
Answer: 20 cm²
Example 2: Example 2: Right-angled triangle
Problem: A right triangle has legs 6 cm and 10 cm. Find its area.
Solution:
- Base = 6 cm, Height = 10 cm (legs are perpendicular).
- Area = 1/2 × 6 × 10 = 30 cm².
Answer: 30 cm²
Example 3: Example 3: Finding the height
Problem: A triangle has area 36 cm² and base 12 cm. Find the height.
Solution:
- Area = 1/2 × base × height
- 36 = 1/2 × 12 × h
- 36 = 6h
- h = 36 ÷ 6 = 6 cm
Answer: Height = 6 cm
Example 4: Example 4: Finding the base
Problem: A triangle has area 45 m² and height 9 m. Find the base.
Solution:
- 45 = 1/2 × b × 9
- 45 = 4.5b
- b = 45 ÷ 4.5 = 10 m
Answer: Base = 10 m
Example 5: Example 5: Triangle from rectangle
Problem: A rectangle is 14 cm × 8 cm. A diagonal divides it into two triangles. Find the area of each triangle.
Solution:
- Rectangle area = 14 × 8 = 112 cm².
- Each triangle = 112 ÷ 2 = 56 cm².
Answer: Each triangle has area 56 cm².
Example 6: Example 6: Comparing two triangles
Problem: Triangle A: base 10 cm, height 6 cm. Triangle B: base 8 cm, height 8 cm. Which has more area?
Solution:
- A = 1/2 × 10 × 6 = 30 cm².
- B = 1/2 × 8 × 8 = 32 cm².
Answer: Triangle B has more area (32 cm² > 30 cm²).
Example 7: Example 7: Triangular garden
Problem: A triangular garden has base 20 m and height 12 m. Find the cost of planting grass at Rs. 5 per m².
Solution:
- Area = 1/2 × 20 × 12 = 120 m².
- Cost = 120 × 5 = Rs. 600.
Answer: Rs. 600
Example 8: Example 8: Using decimals
Problem: Find the area: base = 7.5 cm, height = 4 cm.
Solution:
- Area = 1/2 × 7.5 × 4 = 1/2 × 30 = 15 cm²
Answer: 15 cm²
Example 9: Example 9: Same base and height
Problem: A triangle and a rectangle both have base 10 cm and height 6 cm. Compare their areas.
Solution:
- Rectangle = 10 × 6 = 60 cm².
- Triangle = 1/2 × 10 × 6 = 30 cm².
- Triangle area = half the rectangle area.
Answer: The triangle's area is half of the rectangle's area.
Example 10: Example 10: Word problem
Problem: A triangular plot has base 50 m and height 30 m. It is to be paved at Rs. 80 per m². Find the total cost.
Solution:
- Area = 1/2 × 50 × 30 = 750 m².
- Cost = 750 × 80 = Rs. 60,000.
Answer: Rs. 60,000
Real-World Applications
Construction: Roofs, gable walls, and ramps are often triangular. Builders calculate their area to estimate materials.
Land Measurement: Many plots of land are triangular. Area calculation helps determine the land's value and tax.
Art and Design: Triangles are used in rangoli patterns, logos, and architectural designs. Knowing the area helps plan material usage.
Sails and Flags: Many sails and pennants are triangular. Their area determines the amount of fabric needed.
Science: The cross-section of many objects (prisms, wedges) is a triangle. Its area is needed for volume calculations.
Key Points to Remember
- Area of Triangle = 1/2 × base × height.
- The height must be perpendicular to the base.
- Any side can be chosen as the base; the height changes accordingly.
- A triangle is half of a rectangle with the same base and height.
- For a right triangle, the two legs serve as base and height.
- To find height: h = 2 × Area ÷ base.
- To find base: b = 2 × Area ÷ height.
- Area is always in square units (cm², m²).
Practice Problems
- Find the area: base = 12 cm, height = 7 cm.
- A right triangle has legs 9 cm and 14 cm. Find its area.
- Area is 48 m², base is 16 m. Find the height.
- A rectangle 10 cm × 6 cm is cut along a diagonal. Find the area of each triangle.
- A triangular park has base 40 m and height 25 m. Find the cost of fencing at Rs. 10 per m of perimeter (perimeter = 40 + 30 + 50 m).
- Which has more area: triangle with base 15 cm, height 8 cm, or triangle with base 10 cm, height 12 cm?
- A triangle has area 100 cm². If its base is doubled and height is halved, what is the new area?
- Find the area of a triangle with base 3.5 m and height 4.2 m.
Frequently Asked Questions
Q1. Why is the area of a triangle half of base times height?
Because a triangle is exactly half of a rectangle with the same base and height. If you draw a rectangle and cut it along the diagonal, each half is a triangle.
Q2. What is the height of a triangle?
The height (or altitude) is the perpendicular distance from the base to the opposite vertex. It must make a 90° angle with the base.
Q3. Can any side be the base?
Yes. You can choose any of the three sides as the base. The height will be the perpendicular distance from that side to the opposite vertex. The area will be the same regardless of which side you choose.
Q4. Does this formula work for all triangles?
Yes. The formula Area = 1/2 × base × height works for right, acute, obtuse, scalene, isosceles, and equilateral triangles.
Q5. How do I find the height if it is not given?
If you know the area and base, use: height = 2 × Area ÷ base. In some problems, you may need to measure or calculate the height separately.
Q6. What is the unit of area?
Area is always in square units. If base and height are in cm, area is in cm². If in metres, area is in m².
Q7. Is the height always inside the triangle?
No. For obtuse triangles, the height from the vertex opposite the longest side falls outside the triangle. You extend the base and drop the perpendicular to the extended line.
Q8. How is area different from perimeter?
Area is the space inside the triangle (measured in cm²). Perimeter is the total length around the triangle (measured in cm). They measure different things.
