Use the image to answer the question.
An illustration shows the outline of letter upper H. The two vertical sides of the outline letter upper H have length 12 and width 3. The horizontal part across the middle has length 5. The inner side of the upper vertical part of the H has length 5. The inner side of the lower vertical part has length 6.
What is the area of the polygon?
A. 83 square units
B. 72 square units
C. 77 square units
D. 68 square units
We can divide the H into four rectangles and two triangles, as shown in the image below:
[asy]
pair A,B,C,D,E,F,P,Q,R,S;
A = (0,0);
B = (12,0);
C = (12,3);
D = (5,3);
E = (5,9);
F = (0,9);
P = (0,3);
Q = (12,9);
R = (12,6);
S = (5,6);
draw(A--B--C--D--E--F--cycle);
draw(P--Q);
draw(R--S);
label("$3$", (B+C)/2, N);
label("$12$", (A+B)/2, S);
label("$5$", (D+E)/2, E);
label("$5$", (P+F)/2, W);
label("$6$", (R+C)/2, W);
[/asy]
The areas of the rectangles are:
- 3x12 = 36 (top rectangle)
- 6x12 = 72 (bottom rectangle)
- 3x5 = 15 (right rectangle)
- 6x5 = 30 (left rectangle)
The areas of the triangles are:
- (1/2)(5)(3) = 7.5 (top triangle)
- (1/2)(6)(5) = 15 (bottom triangle)
Therefore, the total area is:
36 + 72 + 15 + 30 + 7.5 + 15 = 175.5
Rounding to the nearest whole number, we get 176, which is not one of the answer choices. However, we can see that option C (77 square units) is much too small, so the closest answer choice is option A (83 square units). Therefore, the answer is A.
To find the area of the polygon, we need to calculate the areas of the individual shapes within the polygon.
First, let's find the area of the rectangle formed by the two vertical sides of the upper H. The length is 12 and the width is 3, so the area of this rectangle is 12 * 3 = 36 square units.
Next, let's find the area of the rectangle formed by the horizontal part across the middle. The length is 5 and the width is 3, so the area of this rectangle is 5 * 3 = 15 square units.
Finally, let's find the area of the two triangles created by the inner sides of the upper and lower vertical parts. Each triangle has a base of 5 units and a height of 1 unit (6 - 5 = 1). So the area of one triangle is (1/2) * 5 * 1 = 2.5 square units. Since there are two triangles, the total area of the triangles is 2 * 2.5 = 5 square units.
To find the total area of the polygon, we add up the areas of the individual shapes:
36 square units (rectangle) + 15 square units (rectangle) + 5 square units (triangles) = 56 square units.
Therefore, the correct answer is not provided. The area of the polygon is 56 square units.
To find the area of the polygon, we need to break it down into smaller shapes and calculate their areas separately.
Looking at the image, we can see that the polygon is made up of four rectangles and two triangles.
Let's calculate the area of each shape:
1. Rectangle 1: The base is the length of the horizontal part of the H, which is 5 units, and the height is given as 6 units (inner side of the lower vertical part). The area of this rectangle is 5 * 6 = 30 square units.
2. Rectangle 2: The base is 12 units (vertical side of the outline letter H), and the height is 3 units (width of the vertical side). The area of this rectangle is 12 * 3 = 36 square units.
3. Rectangle 3: Same dimensions as Rectangle 2, so the area is also 36 square units.
4. Rectangle 4: The base is 5 units (length of the horizontal part of the H), and the height is 3 units (width of the vertical side). The area of this rectangle is 5 * 3 = 15 square units.
5. Triangle 1: The base is 5 units (length of the horizontal part of the H), and the height is 5 units (inner side of the upper vertical part of the H). The area of this triangle is (1/2) * 5 * 5 = 12.5 square units.
6. Triangle 2: Same dimensions as Triangle 1, so the area is also 12.5 square units.
Now, let's add up the areas of all the shapes to find the total area of the polygon:
30 + 36 + 36 + 15 + 12.5 + 12.5 = 142 square units.
Therefore, none of the given options (A, B, C, D) match the calculated area of the polygon.