Learn about how to describe the characteristics of a polygon using mathematical vocabulary, and to reason about the area of polygons. After trying the questions, click on the buttons to view answers and explanations in text or video.
A can be divided into 2 triangles along the black dashed lines. Red dotted lines indicate the height of each triangle. The area of A is (½ × 4 units × 2 units) + (½ × 4 units × 4 units) = 12 square units.
B can be divided into 2 triangles. The area of B is (½ × 4 units × 2 units) + (½ × 4 units × 4 units) = 12 square units.
C can be decomposed and rearranged to make a rectangle. The area of C is 7 units × 4 units = 28 square units.
D can de divided into 2 identical triangles. The area of D is 2 × (½ × 7 units × 2 units) = 14 square units.
E can be divided into a right triangle and a rectangle. The area of E is (½ × 6 units × 3 units) + (2 units × 3 units) = 14 square units.
F can be divided into 2 identical triangles. The area of F is 2 × (½ × 6 units × 3 units) = 18 square units.
Here is a trapezoid. a and b represent the lengths of its bottom and top sides. The segment labeled h represents its height; it is perpendicular to both the top and bottom sides.
Apply area-reasoning strategies—decomposing, rearranging, duplicating, etc.—on the trapezoid so that you have one or more shapes with areas that you already know how to find.
Use the shapes to help you write a formula for the area of a trapezoid. Show your reasoning.
A trapezoid with parallel sides a and b and height h can be duplicated and inverted to create a parallelogram with base a+b and height h.
The area of this new parallelogram is (a + b) · h.
The original trapezoid is half of the parallelogram, so its area A = ½ · (a + b) · h. This is the formula for the area of a trapezoid.
11.4 - Pinwheel
Find the area of the shaded region in square units. Show your reasoning.
The figure can be decomposed into an octagon and four shaded triangles.
The area of the four triangles is 4 × ½ × 2 units × 3 units = 12 square units.
The octagon can be enclosed in a square with area 6 units × 6 units = 36 units.
The area of the four unshaded triangles is 4 × ½ × 2 units × 2 units = 8 square units.
Hence, the area of the octagon is 36 units - 8 units = 28 square units.
The total area of the shaded region is 12 square units + 28 square units = 40 square units.
Lesson 11 Summary
A polygon is a two-dimensional figure composed of straight line segments.
Each end of a line segment connects to one other line segment. The point where two segments connect is a vertex. The plural of vertex is vertices.
The segments are called the edges or sides of the polygon. The sides never cross each other. There are always an equal number of vertices and sides.
Here is a polygon with 5 sides. The vertices are labeled A, B, C, D, and E.
The first two diagrams show the polygon decomposed into triangles and rectangles; the sum of their areas is the area of the polygon. The last diagram shows the polygon enclosed in a rectangle; subtracting the areas of the triangles from the area of the rectangle gives us the area of the polygon.
Glossary Terms
edge
Each straight side of a polygon is called an edge.
A quadrilateral is a type of polygon that has 4 sides. A rectangle is an example of a quadrilateral. A pentagon is not a quadrilateral, because it has 5 sides.
vertex
A vertex is a point where two or more edges meet. When we have more than one vertex, we call them vertices.
A, B, C, and E are polygons.
D is not a polygon because it has line segments with ends that do not meet other line segments; it does not enclose an area.
F is not a polygon because it has line segments which meet multiple other line segments at their ends (because it is a representation of a three-dimensional shape).
2. Mark each vertex with a large dot. How many edges and vertices does this polygon have?
Earlier in this lesson you were asked to find the area of a trapezoid and determine a formula for finding the area of a trapezoid. You may revise that strategy, use the formula, or use another strategy.
Earlier in this lesson, we found that the formula for the area of a trapezoid is A = ½ · (a + b) · h, where a and b are the trapezoid's parallel sides.
Hence, the area of this trapezoid is ½ × (4 units + 8 units) × 3 units = 18 square units.
4. Lin and Andre used different methods to find the area of a regular hexagon with 6-inch sides. Lin decomposed the hexagon into six identical triangles. Andre decomposed the hexagon into a rectangle and two triangles.
Both methods, if done correctly, will give the same answer.
Lin's method: The six triangles are equilateral, so all three sides of any one triangle measure 6 in.
Hence, the height of any one triangle, as shown by the blue dashed line, is 10.4 in ÷ 2 = 5.2 in.
The area of one triangle is ½ × 6 in × 5.2 in = 15.6 sq in.
The area of all six triangles, and hence the area of the hexagon, is 6 × 15.6 sq in = 93.6 sq in.
Andre's method: Note how the two side triangles fit into the middle rectangle. This is a property of a regular hexagon. The height of the triangles is therefore 6 in ÷ 2 = 3 in.
The area of both triangles is 2 × ½ × 10.4 in × 3 in = 31.2 sq in.
The area of the rectangle is 6 in × 10.4 in = 62.4 in.
The area of the hexagon is 31.2 sq in + 62.4 sq in = 93.6 sq in.
5. Identify a base b and a corresponding height h that can be used to find the area of this triangle. Label the base b and the corresponding height h.
Find the area of the triangle. Show your reasoning.
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