Perimeter and Area | A

Question 10

Draw suitable triangles on grid paper to verify your inferences and relationships observed in the above exercises.

Solution

Understand the Question

The area of any triangle drawn on grid paper can be verified using the standard formula: $\text{Area of triangle} = \dfrac{1}{2} \times \text{base} \times \text{height}$
We verify this relationship for two cases:
1. Right-angled triangle: It occupies exactly half the area of an enclosing rectangle of the same base and height.
2. General (non-right) triangle: It can be split along its height into two right-angled triangles whose individual areas add up to the total area.

Step 1 · Verify Area of a Right-Angled Triangle

Draw a right-angled triangle with $\text{base} = 6\text{ units}$ and $\text{height} = 4\text{ units}$ enclosed in a $6 \times 4$ rectangle. Diagram 1

\begin{aligned} \text{Area of rectangle} &= \text{base} \times \text{height} \\[0.6em] &= 6 \times 4 \\[0.6em] &= 24 \text{ square units} \end{aligned}

Since the triangle is half of the rectangle

\begin{aligned} \text{Area of right-angled triangle} &= \dfrac{1}{2} \times \text{Area of rectangle} \\[0.6em] &= \dfrac{1}{2} \times 24 \\[0.6em] &= 12 \text{ square units} \end{aligned}

Using the triangle area formula

\begin{aligned} \text{Area of triangle} &= \dfrac{1}{2} \times \text{base} \times \text{height} \\[0.6em] &= \dfrac{1}{2} \times 6 \times 4 \\[0.6em] &= \dfrac{1}{2} \times 24 \\[0.6em] &= 12 \text{ square units} \end{aligned}

The grid counting and formula results match.

Step 2 · Verify Area of a General Triangle

Draw a general triangle with $\text{base} = 8\text{ units}$ and $\text{height} = 5\text{ units}$ . Diagram 2

Using the triangle area formula

\begin{aligned} \text{Area of triangle} &= \dfrac{1}{2} \times \text{base} \times \text{height} \\[0.6em] &= \dfrac{1}{2} \times 8 \times 5 \\[0.6em] &= \dfrac{1}{2} \times 40 \\[0.6em] &= 20 \text{ square units} \end{aligned}

Splitting the triangle into two right-angled triangles along the altitude

\begin{aligned} \text{Area of first triangle} &= \dfrac{1}{2} \times 2 \times 5 = 5 \text{ square units} \\[0.6em] \text{Area of second triangle} &= \dfrac{1}{2} \times 6 \times 5 = 15 \text{ square units} \\[0.6em] \text{Total Area} &= 5 + 15 = 20 \text{ square units} \end{aligned}

The sum of parts matches the formula result.

Answer

The area formula $\text{Area} = \dfrac{1}{2} \times \text{base} \times \text{height}$ is verified for both right-angled and general triangles on grid paper.

Common Mistakes

Slant Height vs. Perpendicular Height: Using the length of a slanted side instead of the perpendicular height from the vertex to the base line.
Counting Partial Squares Inaccurately: Miscounting fractional grid squares instead of splitting the figure into well-defined right-angled triangles.

Question 10

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