Solved Exercises on Area of Triangles in Grade 3

Master area of triangles: base, height, formula, calculations with detailed explanations through these 5 exercises.

Solution: Exercises 1 to 3
1 Basic triangle area
Exercise 1
Find the area of a triangle with base 6 cm and height 4 cm.
Definition:

Triangle: A shape with 3 sides and 3 angles

Base: The bottom side of the triangle

Height: The vertical distance from base to opposite vertex

Area calculation method:
  1. Identify the base and height of the triangle
  2. Use the formula: Area = (base × height) ÷ 2
  3. Calculate the multiplication first
  4. Divide by 2 to get the final area
Formula
Area = (b × h) ÷ 2
Values
b = 6, h = 4
Result
12 cm²
Step 1: Identify the measurements

Base = 6 cm, Height = 4 cm

Step 2: Write the formula

Area of triangle = (base × height) ÷ 2

Step 3: Substitute the values

Area = (6 × 4) ÷ 2

Step 4: Calculate the multiplication

6 × 4 = 24

Step 5: Divide by 2

24 ÷ 2 = 12

Area = 12 cm²
Final answer:

The area of the triangle is 12 square centimeters.

Applied rules:

• Triangle Area Formula: Area = (base × height) ÷ 2

• Multiplication: Calculate base × height first

• Division: Divide the product by 2

• Units: Area is measured in square units (cm²)

Tip 1: Always identify base and height before calculating.
Tip 2: Remember to divide by 2 - triangles are half of rectangles!
2 Triangle with different units
Exercise 2
Find the area of a triangle with base 8 inches and height 5 inches.
Definition:

Area: The amount of space inside a shape

Square inch: A unit of area equal to a square with sides of 1 inch

Formula
Area = (b × h) ÷ 2
Values
b = 8, h = 5
Result
20 in²
Step 1: Identify the measurements

Base = 8 inches, Height = 5 inches

Step 2: Write the formula

Area of triangle = (base × height) ÷ 2

Step 3: Substitute the values

Area = (8 × 5) ÷ 2

Step 4: Calculate the multiplication

8 × 5 = 40

Step 5: Divide by 2

40 ÷ 2 = 20

Area = 20 in²
Final answer:

The area of the triangle is 20 square inches.

Applied rules:

• Consistent Units: Both base and height must be in the same unit

• Area Formula: Area = (base × height) ÷ 2

• Division: Always divide by 2 for triangle area

• Unit Squared: Area units are always squared (in², cm², etc.)

Tip 1: Check that both measurements are in the same unit.
Tip 2: Don't forget to write the answer in square units!
3 Finding missing height
Exercise 3
A triangle has an area of 15 cm² and a base of 6 cm. What is its height?
Definition:

Missing measurement: When one value is unknown in the area formula

Algebra: Using the area formula backwards to find unknown values

Formula
Area = (b × h) ÷ 2
Known Values
Area = 15, b = 6
Find
h = ?
Step 1: Write the formula

Area = (base × height) ÷ 2

Step 2: Substitute known values

15 = (6 × height) ÷ 2

Step 3: Multiply both sides by 2

15 × 2 = 6 × height

30 = 6 × height

Step 4: Divide both sides by 6

30 ÷ 6 = height

height = 5

Step 5: Verify the answer

Area = (6 × 5) ÷ 2 = 30 ÷ 2 = 15 ✓

Height = 5 cm
Final answer:

The height of the triangle is 5 cm.

Applied rules:

• Algebra: Rearrange the formula to solve for unknown values

• Balance: Whatever you do to one side, do to the other

• Verification: Check your answer by substituting back

• Area Formula: Area = (base × height) ÷ 2

Tip 1: When finding missing values, rearrange the formula.
Tip 2: Always verify your answer by substituting back.
Triangle Area Rules and Methods
Area = (base × height) ÷ 2
Triangle Area Formula
Triangle
3 sides, 3 angles
Basic geometric shape
Base
Bottom side
Length of bottom edge
Height
Vertical distance
Perpendicular to base
Key definitions:

Triangle: A polygon with 3 straight sides and 3 angles

Base: Any side of the triangle that serves as the bottom

Height: The perpendicular distance from base to opposite vertex

Area: The amount of space inside the triangle

Complete methodology:
  1. Identify base and height: Find the measurements given
  2. Write the formula: Area = (base × height) ÷ 2
  3. Substitute values: Replace variables with numbers
  4. Calculate: Multiply first, then divide by 2
  5. Check units: Answer should be in square units
Tip 1: The height is always perpendicular to the base.
Tip 2: Triangles are half the area of their corresponding rectangle.
Tip 3: Always include square units in your final answer.
Tip 4: If one measurement is missing, rearrange the formula.
Common errors: Forgetting to divide by 2, mixing up base and height, wrong units.
Memory trick: Think "triangle is half of rectangle" - that's why we divide by 2.
Formulas to know by heart:

• Triangle Area: Area = (base × height) ÷ 2

• Rectangle Area: Area = length × width

• Triangle relationship: Triangle = ½ Rectangle

• Unit conversion: cm², in², ft², etc.

Solution: Exercises 4 to 5
4 Real-world triangle
Exercise 4
Sarah wants to paint a triangular wall with base 12 feet and height 8 feet. How much area does she need to paint?
Definition:

Real-world problem: Applying math to practical situations

Area calculation: Finding how much space a shape covers

Given
b = 12 ft, h = 8 ft
Formula
Area = (b × h) ÷ 2
Result
48 ft²
Step 1: Identify the measurements

Base = 12 feet, Height = 8 feet

Step 2: Write the formula

Area of triangle = (base × height) ÷ 2

Step 3: Substitute the values

Area = (12 × 8) ÷ 2

Step 4: Calculate the multiplication

12 × 8 = 96

Step 5: Divide by 2

96 ÷ 2 = 48

Step 6: State the answer

Sarah needs to paint 48 square feet

Area = 48 ft²
Final answer:

Sarah needs to paint 48 square feet of wall.

Applied rules:

• Problem solving: Extract relevant information from word problems

• Area formula: Area = (base × height) ÷ 2

• Real-world context: Connect math to practical applications

• Units: Use correct units (feet → square feet)

Tip 1: Draw a picture to visualize the problem.
Tip 2: Circle important numbers in word problems.
5 Comparing triangles
Exercise 5
Triangle A has base 10 cm and height 6 cm. Triangle B has base 8 cm and height 7 cm. Which triangle has the larger area?
Definition:

Comparison: Finding which value is greater or lesser

Multiple calculations: Solving more than one problem

Triangle A
(10 × 6) ÷ 2 = 30
Triangle B
(8 × 7) ÷ 2 = 28
Comparison
30 > 28
Step 1: Calculate Triangle A's area

Area = (10 × 6) ÷ 2 = 60 ÷ 2 = 30 cm²

Step 2: Calculate Triangle B's area

Area = (8 × 7) ÷ 2 = 56 ÷ 2 = 28 cm²

Step 3: Compare the areas

30 cm² vs 28 cm²

Step 4: Determine which is larger

30 > 28, so Triangle A has the larger area

Step 5: State the conclusion

Triangle A has the larger area

Triangle A has the larger area (30 cm² > 28 cm²)
Final answer:

Triangle A has the larger area (30 cm² compared to 28 cm²).

Applied rules:

• Comparison: Calculate each area separately, then compare

• Multiple calculations: Solve each triangle individually

• Area formula: Area = (base × height) ÷ 2

• Greater than: Use > symbol to show comparison

Tip 1: Organize your work when comparing multiple shapes.
Tip 2: Always label which calculation belongs to which shape.
Triangle Area Laws, Methods, Rules, and Definitions
Area = (base × height) ÷ 2
Triangle Area Formula
Key definitions:

Triangle: A polygon with 3 straight sides and 3 angles

Base: Any side of the triangle used as the reference side

Height: The perpendicular distance from the base to the opposite vertex

Area: The amount of space inside a 2D shape

Vertex: A corner where two sides meet

Perpendicular: At a 90-degree angle

Complete methodology:
  1. Identify the triangle: Recognize the shape as a triangle
  2. Find the base: Locate the bottom side of the triangle
  3. Find the height: Measure the perpendicular distance from base to top
  4. Apply the formula: Area = (base × height) ÷ 2
  5. Calculate: Multiply base by height, then divide by 2
  6. Check units: Express answer in square units
Tip 1: The base can be any side of the triangle.
Tip 2: The height must be perpendicular to the base.
Tip 3: Triangles take up half the space of their corresponding rectangle.
Tip 4: Always include square units in your final answer.
Common errors: Forgetting to divide by 2, using slanted height instead of perpendicular height, wrong units.
Memory aid: "Triangle is half of rectangle" - that's why we divide by 2.
Formulas to know by heart:

• Triangle Area: Area = (base × height) ÷ 2

• Rectangle Area: Area = length × width

• Triangle relationship: Triangle Area = ½ × Rectangle Area

• Unit conversion: cm², in², ft², m², etc.

Triangle
3 sides, 3 angles
Basic geometric shape
Base
Bottom side
Reference measurement
Height
Perpendicular distance
From base to vertex
Area
Space inside
Measured in squares

Questions & Answers

Question: Why do we divide by 2 when calculating the area of a triangle? It seems like we're making the number smaller.

Answer: Great question! We divide by 2 because a triangle is exactly half the area of a rectangle with the same base and height.

Think of it this way: If you draw a rectangle and cut it diagonally from corner to corner, you get two identical triangles. Each triangle takes up exactly half of the rectangle's space.

For example, if you have a rectangle that is 6 cm wide and 4 cm tall, its area is 6 × 4 = 24 cm². If you cut this rectangle diagonally, each triangle would have an area of 24 ÷ 2 = 12 cm².

That's why the triangle area formula is: Area = (base × height) ÷ 2. The division by 2 accounts for the fact that triangles are half of rectangles!

Question: In some triangles, the height doesn't seem to go straight down from the top. How do I measure the height then?

Answer: The height of a triangle is always measured as the perpendicular distance from the base to the opposite vertex (corner). This means it forms a perfect L-shape with the base.

Even if the triangle looks "slanted" or "lopsided," you still measure the height straight up or down to make a 90-degree angle with the base. Sometimes you might need to extend the base line to measure the height properly.

For example, in an obtuse triangle (where one angle is bigger than 90 degrees), the height might fall outside the triangle itself. But you still measure it as the shortest distance from the base to the opposite vertex, making sure it's perfectly vertical (perpendicular) to the base.

Remember: Height is always perpendicular to the base, regardless of how the triangle is positioned!

Question: Does it matter which side I choose as the base of the triangle? Will I get a different answer?

Answer: No, it doesn't matter which side you choose as the base! You will always get the same area answer regardless of which side you select as the base.

However, when you change the base, you must also change the corresponding height. The height is always measured from the chosen base to the opposite vertex.

For example, if you have a triangle with sides of 3 cm, 4 cm, and 5 cm, you could:

  • Use the 3 cm side as the base and measure its corresponding height
  • Use the 4 cm side as the base and measure its corresponding height
  • Use the 5 cm side as the base and measure its corresponding height

In each case, when you calculate (base × height) ÷ 2, you'll get the exact same area. This is a fundamental property of triangles!