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Shape Up

Problem

Determine the area of each of the four shapes below. In order to find the area, you may divide each shape up into other shapes in any way you need to.

Note: All linear measures are in cm. Shape 1 has a vertical line of symmetry, Shape 2 has a horizontal line of symmetry, and Shape 3 has both a horizontal and a vertical line of symmetry. All angles in Shape 3 are right angles.

Shape 1 is a four sided polygon. It has two horizontal sides with top side of length 5 and bottom side of length 7. The distance between the top and bottom sides is 4 units. Shape 2 is a composite shape made up of a square of side length 4 and a triangle. The base of the triangle runs vertically and is placed along one side of the square making a shape that looks like an arrow pointing right. The base of the triangle extends beyond the square by 2 units above and 2 units below. The height of the triangle, which runs horizontally, is 4 units.

Shape 3 is a composite shape made up of three side by side rectangles: The middle rectangle has horizontal sides of length 8 and vertical sides of length 4. The left and right rectangles are identical. The top side of the left rectangle has length 1 and is 1 unit below the top side of the middle rectangle. Shape 4 is a four sided polygon with two vertical sides of length 4 and 5 and one horizontal side of length 8.

Solution

Shape 1:

Solution 1: Divide the shape into two right-angled triangles and a rectangle, as shown.

A vertical line is drawn from each vertex of the top side of length 5 to the bottom side of length 7. These lines meet the bottom side at right angles and divide the bottom side into pieces of length 1, 5, and 1.

Since there a vertical line of symmetry, the base of each triangle is \(\frac{7-5}{2}=1.\) So each triangle has area \(\frac{1\times 4}{2} = 2\).
The rectangle has dimensions 5 by 4, and so its area is \(5\times 4 = 20\).

Therefore, the area of Shape 1 is \(2 + 2 + 20 = 24\mbox{ cm}^2\).

Solution 2: Divide the shape into two triangles, as shown.

A diagonal line is drawn from top right vertex to the bottom left vertex. The heights of the two triangles formed are equal to height of Shape 1 which is 4.

The bottom triangle has a base of 7 and a height of 4, so the area of this triangle is \(\frac{7\times 4}{2} = 14\).
The top triangle has a base of 5 and a height of 4. (You may have to turn the page upside down to see this.) So the area of this triangle is \(\frac{5\times 4}{2} = 10\).

Therefore, the area of Shape 1 is \(14 + 10 = 24\mbox{ cm}^2\).

Shape 2:

Divide the shape into a square and a triangle, as shown.

Shape 2 is a composite shape that looks like an arrow pointing right. A vertical line of length 4 divides the arrow into two pieces.

The square has a side length of 4, so the area of the square is \(4\times 4 = 16\).
The triangle has base \(2+4+2=8\) and height 4, so the area of the triangle is \(\frac{8\times 4}{2} = 16\).

Therefore, the area of Shape 2 is \(16 + 16 = 32\mbox{ cm}^2\).

Shape 3:

Solution 1: Divide the shape into three rectangles, vertically, as shown to the right.

Shape 3 is a composite shape made up of three side by side rectangles. Two vertical lines divide the shape into three pieces: a middle rectangle and two identical side rectangles.

Each side rectangle has length \(4 -1 -1 =2\) and width 1, so the area of each side rectangle is \(2\times 1 = 2\).
The middle rectangle has the dimensions of 8 by 4, so its area is \(8\times 4 = 32\).

Therefore, the area of Shape 3 is \(2 + 2 + 32 = 36\mbox{ cm}^2\).

Solution 2: Divide the shape into three rectangles, horizontally, as shown.

Two horizontal lines divide the shape into three pieces: two identical top and bottom rectangles with vertical sides of length 1, and a middle rectangle that extends past the other rectangles by 1 unit on the left and 1 unit the right.

The top and bottom rectangles have the dimensions of 1 by 8, so the area of each is \(1\times 8 = 8\).
The length of the middle rectangle is \(8+ 1 + 1 = 10\) and its width is \(4 - 1- 1 = 2\), so the middle rectangle has area \(10\times 2 = 20\).

Therefore, the area of Shape 3 is \(8 + 8 + 20= 36\mbox{ cm}^2\).

Solution 3: Divide the shape into one rectangle with four 1 by 1 corners removed, as shown.

Shape 3 is a large rectangle with a 1 by 1 square removed from each corner.

The large rectangle has a length \(8 + 1 + 1 = 10\) and width is 4, so its area is \(10\times 4 = 40\). Each of the corner squares has dimensions of 1 by 1, so area \(1\times 1 = 1\).

Therefore, the area of Shape 3 is \(40 - 4 \times 1 = 36\mbox{ cm}^2\).

Shape 4:

Solution 1: Divide the shape into a rectangle and a triangle, as shown.

A horizontal line is drawn through Shape 4 from the top of the side of length 4 over to the side of length 5. This line divides the shape into a rectangle of height 4 and a right-angled triangle of height 1. The base of the triangle lies along the top side of the rectangle.

The base of the triangle is 8 and the height is \(5 - 4 = 1\), so the area of the triangle is \(\frac{8\times 1}{2} = 4\). The rectangle has the dimensions 4 by 8, so area \(4\times 8 = 32\).
Therefore, the area of Shape 4 is \(4 + 32 = 36\mbox{ cm}^2\).

Solution 2: Divide the shape into two triangles, as shown.

A diagonal line is drawn from the top left vertex to the bottom right vertex of Shape 4. This divides the shape into a lower triangle with base lying along the bottom edge of the shape and an upper triangle with base running vertically along the right edge of the shape.

The lower triangle has a base of 8 and a height of 4, so the area of the triangle is \(\frac{8\times 4}{2} = 16\). The upper triangle has a base of 5 and a height of 8, so the area of the triangle is \(\frac{5\times 8}{2} = 20\). Therefore, the area of Shape 4 is \(16 + 20 = 36\mbox{ cm}^2\).