Math Labs with Activity - Derive a Formula for Finding the Area of a Triangle

Math Labs with Activity – Derive a Formula for Finding the Area of a Triangle

OBJECTIVE

To derive a formula for finding the area of a triangle

Materials Required

1. Three sheets of white paper
2. A sheet of glazed paper
3. A geometry box
4. A tube of glue
5. A pair of scissors

Theory
It has been geometrically proved that the area of a triangle having base =b units and height = h units is given by square units.

Procedure
Step 1: We shall first consider an acute-angled triangle.
Step 2: Construct an acute-angled triangle on a sheet of white paper as shown in Figure 27.1(a).

Step 3: Make an exact copy of this triangle on the glazed paper. From the upper vertex drop a perpendicular to the opposite side as shown in Figure 27.1(b). Cut the triangle along its boundary. Also, cut it into two parts along the perpendicular.
Step 4: Paste these two triangles adjacent to ΔABC on the white paper so as to form a rectangle as shown in Figure 27.1(c).
Step 5: Record your observations (see Observation 1).
Step 6: We shall now consider a right-angled triangle.
Step 7: Construct a right-angled triangle on another sheet of white paper as shown in Figure 27.2(a).
Step 8: Make an exact copy of this triangle on the glazed paper. Cut this triangle along its boundary and paste it adjacent to the triangle on the white paper so as to form a rectangle as shown in Figure 27.2(b).
Step 9: Record your observations (see Observation 2).
Step 10: We shall now consider an obtuse-angled triangle.
Step 11: Construct an obtuse-angled triangle on the third sheet of white paper as shown in Figure 27.3(a).

Step 12: Make an exact copy of this triangle on the glazed paper. Cut this triangle along its boundary and paste it adjacent to the triangle on the white paper so as to form a parallelogram as shown in Figure 27.3(b).
Step 13: Record your observations (see Observation 3).

Observations

1. The area of the rectangle formed is equal to the total area of the two congruent triangles.
   ∴ area of the triangle = ½ x area of the rectangle
   = ½ x length x breath
   = ½ x h x b.
   [∴ length of the rectangle = height of the triangle = b, and breadth of the rectangle = height of the triangle = h]
2. The area of the rectangle formed is equal to the total area of the two congruent triangles.
   ∴ area of the triangle=½ x area of the rectangle
   = ½ x length x breadth
   = ½ x h x b.
   [∴ length of the rectangle = height of the triangle = h, and breadth of the rectangle = base of the triangle =b]
3. The area of the parallelogram formed is equal to the total area of the two congruent triangles.
   ∴ area of the triangle = ½ x area of the parallelogram
   = ½ x base x height.
   = ½ x b x h.
   [‍∴ base of the parallelogram = base of the triangle =b, and height of the parallelogram = height of the triangle = h]

Result
The area of a triangle is half the product of its base and height.

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