Area of Triangles Section 8.4

Goal Find the area of triangles.

Key Vocabulary Height of a triangle Base of a triangle

Theorem 8.3 Areas of Similar Polygons

The Area Of A Triangle. Consider the right angled triangle below: What shape is the triangle half of ? Rectangle 8 cm 5cm What is the area of the rectangle? Area = 8 x 5 = 40 cm 2 What is the area of the triangle ? Area = ½ x 40 = 20cm 2 base height The formula for the area of a triangle is: Area = ½ x base x height A = ½ bh

The altitude, or height, of a triangle is the perpendicular distance from either the base of the triangle or an extension of the base to the opposite vertex. Area of Triangles h b h b h b Area of a Triangle Very Important: The height must be perpendicular to the base.

Does the formula apply to all triangles ? base (b) height (h) Can we make this triangle into a rectangle ? Yes The triangle is half the area of this rectangle: B H A1 The areas marked A1 are equal. A2 The areas marked A2 are equal. For all triangles: Area = ½ bh

Triangles With The Same Area Triangles can have the same area without necessarily being congruent. For example, all the triangles below have the same area but they are not congruent

Calculate the areas of the triangles below: Example 1 10cm 6cm Solution. Area = ½ x base x height base = 10 cm height = 6cm Area = ½ x 10 x 6 Area = ½ x 60 = 30cm 2 Example 2 6.4m 3.2m Solution. Area = ½ x base x height base = 6.4m height = 3.2m Area = ½ x 6.4 x 3.2 Area = ½ x = 10.24m 2

Example 3 Find the Area of a Right Triangle Find the area of the right triangle. SOLUTION Use the formula for the area of a triangle. Substitute 10 for b and 6 for h. = 1 2 (10)(6) Substitute 10 for b and 6 for h. = 30 Simplify. ANSWER The triangle has an area of 30 square centimeters. Formula for the area of a triangle A = bhbh 1 2

Example 4 Find the Area of a Triangle = 1 2 (8)(5) Substitute 8 for b and 5 for h. = 20 Simplify. ANSWER The triangle has an area of 20 square feet. Find the area of the triangle. SOLUTION Formula for the area of a triangle A = bhbh 1 2

Example 5 Find the Height of a Triangle 78 = 13h Multiply each side by 2. ANSWER The triangle has a height of 6 inches. Find the height of the triangle, given that its area is 39 square inches. Substitute 39 for A and 13 for b. 1 2 (13)h 39 = 6 = h Divide each side by 13. SOLUTION Formula for the area of a triangle A = bhbh 1 2

Your Turn: ANSWER 36 in. 2 In Exercises 1–3, find the area of the triangle. ANSWER 42 yd 2 ANSWER 12 in A triangle has an area of 84 square inches and a height of 14 inches. Find the base. ANSWER 120 cm 2

Example 6 16m 12m 10m Calculate the area of the shape below:Solution. Divide the shape into parts: A1 A2 Area = A1 + A2 A1 A =4 10 Area = bh + 1/2 bh Area = 10 x 12 + ½ x 4 x 10 Area = Area = 140m 2

Area of Similar Triangles The similarity ratio from triangle #1 to #2 is ½. We can use the formula for the area of a triangle to find that Area of Triangle #1 = ½(12 4) = 24 Area of Triangle #2 = ½(24 8) = 96 As you can see from our knowledge of the formula of area, the ratio of the areas is

Area of Similar Triangles To find the ratio of the areas of similar triangles, just square the similarity ratio. The relationship for the ratio of areas of similar triangles can be generalized for all similar polygons.

Areas of Similar Polygons

Example 6 Areas of Similar Triangles = Ratio of areas = Area of ∆ABC Area of ∆DEF 4 9 Find the ratio of the areas of the similar triangles. a. Find the scale factor of ∆ABC to ∆DEF and compare it to the ratio of their areas. b. SOLUTION a. Area of∆ABC = 1 2 bh = 1 2 (4)(2) = 4 square units Area of∆DEF = 1 2 bh = 1 2 (6)(3)= 9 square units

Example 6 Areas of Similar Triangles b. The scale factor of  ABC to  DEF is. 2 3 The ratio of the areas is the square of the scale factor:. =

Assignment Pg. 434 – 437: #1 – 37 odd