Properties of Area

New York State Common Core Math Geometry, Module 3, Lesson 2

Worksheets for Geometry

Student Outcomes

Students understand properties of area:

1. Students understand that the area of a set in the plane is a number greater than or equal to zero that measures the size of the set and not the shape.
2. The area of a rectangle is given by the formula length × width. The area of a triangle is given by the formula 1/2 base × height. A polygonal region is the union of finitely many non-overlapping triangular regions and has area the sum of the areas of the triangles.
3. Congruent regions have the same area.
4. The area of the union of two regions is the sum of the areas minus the area of the intersection.
5. The area of the difference of two regions where one is contained in the other is the difference of the areas.

Properties of Area

Classwork

Exploratory Challenge/Exercises 1–4

1. Two congruent triangles are shown below
   a. Calculate the area of each triangle.
   b. Circle the transformations that, if applied to the first triangle, would always result in a new triangle with the same area:
   Translation Rotation Dilation Reflection
   c. Explain your answer to part (b).

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2. a. Calculate the area of the shaded figure below.
   b. Explain how you determined the area of the figure.
3. Two triangles △ 𝐴𝐵𝐶 and △ 𝐷𝐸𝐹 are shown below. The two triangles overlap forming △ 𝐷𝐺𝐶
   a. The base of figure 𝐴𝐵𝐺𝐸𝐹 is composed of segments of the following lengths: 𝐴𝐷 = 4, 𝐷𝐶 = 3, and 𝐶𝐹 = 2. Calculate the area of the figure 𝐴𝐵𝐺𝐸𝐹.
   b. Explain how you determined the area of the figure.
4. A rectangle with dimensions 21.6 × 12 has a right triangle with a base 9.6 and a height of 7.2 cut out of the rectangle.
   a. Find the area of the shaded region.
   b. Explain how you determined the area of the shaded region.

Lesson Summary

SET (description): A set is a well-defined collection of objects called elements or members of the set.

SUBSET: A set 𝐴 is a subset of a set 𝐵 if every element of 𝐴 is also an element of 𝐵. The notation 𝐴 ⊆ 𝐵 indicates that the set 𝐴 is a subset of set 𝐵.

UNION: The union of 𝐴 and 𝐵 is the set of all objects that are either elements of 𝐴 or of 𝐵, or of both. The union is denoted 𝐴 ∪ 𝐵.

INTERSECTION: The intersection of 𝐴 and 𝐵 is the set of all objects that are elements of 𝐴 and also elements of 𝐵. The intersection is denoted 𝐴 ∩ 𝐵.

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