Area of Triangle (Grade 5)
You already know how to find the area of a rectangle. A triangle is exactly half of a rectangle (or parallelogram) when you draw a diagonal. This gives us a simple formula for the area of a triangle.
Triangles are among the most important shapes in geometry. They appear in rooftops, bridges, sails, road signs, and decorations. Knowing how to calculate their area is essential for measurement, construction, and design.
In Class 5, you will learn the formula for the area of a triangle, understand why it works, practise identifying the base and height, and apply the formula to solve real-life problems.
What is Area of Triangle - Class 5 Maths (Measurement)?
The area of a triangle is the amount of surface enclosed by its three sides.
Every triangle has a base and a corresponding height (altitude). The height is the perpendicular distance from the base to the opposite vertex.
Area of Triangle = ½ × Base × Height
Why does this formula work?
A rectangle has area = base × height. A diagonal divides the rectangle into two equal triangles. So each triangle has area = half the rectangle = ½ × base × height.
Area of Triangle (Grade 5) Formula
Area of Triangle = ½ × b × h
where b = base and h = height (perpendicular to the base).
Important: The height must be perpendicular (at 90°) to the chosen base. In a right triangle, the two shorter sides (legs) serve as base and height.
Types and Properties
Finding the height for different triangles:
- Right triangle: The two legs are the base and height. Easy to calculate.
- Acute triangle: Drop a perpendicular from the top vertex to the base. The height falls inside the triangle.
- Obtuse triangle: The height may fall outside the triangle (you extend the base line and drop a perpendicular to it).
Any side of a triangle can be chosen as the base. The height changes accordingly, but the area remains the same.
Solved Examples
Example 1: Example 1: Basic Area Calculation
Problem: Find the area of a triangle with base 10 cm and height 6 cm.
Solution:
Step 1: Area = ½ × base × height
Step 2: Area = ½ × 10 × 6 = ½ × 60 = 30 cm²
Answer: The area is 30 cm².
Example 2: Example 2: Right Triangle
Problem: A right triangle has legs of 8 cm and 5 cm. Find its area.
Solution:
Step 1: In a right triangle, the two legs serve as base and height.
Step 2: Area = ½ × 8 × 5 = ½ × 40 = 20 cm²
Answer: The area is 20 cm².
Example 3: Example 3: Finding the Height
Problem: A triangle has area 54 cm² and base 12 cm. Find the height.
Solution:
Step 1: Area = ½ × base × height
Step 2: 54 = ½ × 12 × height
Step 3: 54 = 6 × height
Step 4: height = 54 ÷ 6 = 9 cm
Answer: The height is 9 cm.
Example 4: Example 4: Finding the Base
Problem: The area of a triangle is 35 cm² and height is 7 cm. Find the base.
Solution:
Step 1: 35 = ½ × base × 7
Step 2: 35 = 3.5 × base
Step 3: base = 35 ÷ 3.5 = 10 cm
Answer: The base is 10 cm.
Example 5: Example 5: Word Problem - Triangular Garden
Problem: Priya has a triangular flower bed with base 8 m and height 5 m. She wants to spread fertiliser that costs ₹12 per sq m. Find the cost.
Solution:
Step 1: Area = ½ × 8 × 5 = 20 m²
Step 2: Cost = 20 × 12 = ₹240
Answer: The cost is ₹240.
Example 6: Example 6: Triangle Inside a Rectangle
Problem: A rectangle is 14 cm × 8 cm. A diagonal divides it into two triangles. Find the area of each triangle.
Solution:
Step 1: Area of rectangle = 14 × 8 = 112 cm²
Step 2: The diagonal divides the rectangle into two equal triangles.
Step 3: Area of each triangle = 112 ÷ 2 = 56 cm²
Answer: Each triangle has area 56 cm².
Example 7: Example 7: Comparing Areas
Problem: Triangle A has base 12 cm and height 10 cm. Triangle B has base 15 cm and height 8 cm. Which has more area?
Solution:
Step 1: Area of A = ½ × 12 × 10 = 60 cm²
Step 2: Area of B = ½ × 15 × 8 = 60 cm²
Answer: Both triangles have equal area (60 cm²).
Example 8: Example 8: Decimal Measurements
Problem: Find the area of a triangle with base 7.5 cm and height 4 cm.
Solution:
Step 1: Area = ½ × 7.5 × 4 = ½ × 30 = 15 cm²
Answer: The area is 15 cm².
Example 9: Example 9: Area of a Sail
Problem: A triangular sail has a base of 3 m and a height of 4.5 m. Find its area.
Solution:
Step 1: Area = ½ × 3 × 4.5 = ½ × 13.5 = 6.75 m²
Answer: The area is 6.75 m².
Example 10: Example 10: Double the Base
Problem: If the base of a triangle is doubled while the height stays the same, what happens to the area?
Solution:
Step 1: Original area = ½ × b × h
Step 2: New area = ½ × 2b × h = 2 × (½ × b × h) = 2 × original area
Answer: The area doubles.
Real-World Applications
Where do we use the area of triangles?
- Land measurement: Surveyors divide irregular plots into triangles and calculate each triangle’s area to find the total land area.
- Roofing: Triangular roof sections (gable roofs) need material calculated using the triangle area formula.
- Sails and flags: Triangular sails and pennant flags are measured by base and height.
- Art and design: Triangular patterns in rangoli, quilts, and floor designs use area calculations for proper sizing.
- Engineering: Triangular cross-sections in bridges and trusses are analysed using area formulas.
Key Points to Remember
- Area of a triangle = ½ × base × height.
- The height must be perpendicular to the chosen base.
- In a right triangle, the two legs are the base and height.
- A triangle is half of the rectangle (or parallelogram) with the same base and height.
- Any side can be chosen as the base; the height changes accordingly, but the area remains the same.
- If the base is doubled (height unchanged), the area doubles. If both are doubled, the area becomes 4 times.
- Area is measured in square units (cm², m²).
Practice Problems
- Find the area of a triangle with base 14 cm and height 9 cm.
- A right triangle has legs 6 cm and 8 cm. Calculate its area.
- The area of a triangle is 42 cm² and the base is 14 cm. Find the height.
- A triangular plot has base 20 m and height 15 m. Find its area in square metres.
- A rectangle is 10 cm × 6 cm. What is the area of each triangle formed by its diagonal?
- Dev doubles the height of a triangle while keeping the base the same. By how much does the area increase?
- A triangle has base 9.5 cm and height 4 cm. Find its area.
- The area of a triangular banner is 36 m². The base is 12 m. How tall is the banner?
Frequently Asked Questions
Q1. What is the formula for the area of a triangle?
Area of a triangle = ½ × base × height, where the height is perpendicular to the base.
Q2. Why is it half of base times height?
A rectangle with the same base and height has area = base × height. A diagonal cuts the rectangle into two equal triangles. So each triangle = half the rectangle.
Q3. What is the height of a triangle?
The height (or altitude) is the perpendicular distance from the base to the opposite vertex. It forms a 90° angle with the base.
Q4. Can any side be the base?
Yes. You can choose any side as the base. The height will then be the perpendicular distance from that base to the opposite vertex. The area will be the same regardless of which side you choose.
Q5. How do you find the height of a right triangle?
In a right triangle, the two sides forming the right angle (legs) serve as base and height. No additional construction is needed.
Q6. What happens to the area if both base and height are doubled?
The area becomes 4 times the original. New area = ½ × 2b × 2h = 4 × (½ × b × h).
Q7. What units is the area of a triangle measured in?
Area is measured in square units: cm² (square centimetres), m² (square metres), km² (square kilometres), etc.
Q8. Can two triangles with different shapes have the same area?
Yes. Any two triangles with the same base and the same height have the same area, even if they look completely different.
Q9. How is the area of a triangle related to the area of a parallelogram?
A triangle with a given base and height has exactly half the area of a parallelogram with the same base and height.
Q10. Is this topic in the NCERT Class 5 syllabus?
Yes. The area of a triangle is part of the Measurement chapter in NCERT/CBSE Class 5 Maths.
