Question

what is the area of this figure?
17 mm
3 mm
5 mm
4 mm
9 mm
3 mm
2 mm
9 mm
square millimeters
submit
work it out

Explanation:

Step1: Divide the figure into parts

We can divide the composite figure into three rectangles.
- Rectangle 1: Length = 17 mm, Width = 3 mm. Area formula: \( A = l \times w \), so \( A_1 = 17 \times 3 \).
- Rectangle 2: Length = \( 17 - 5 = 12 \) mm? Wait, no, let's re - examine. Wait, another way: Let's find the dimensions of each part.
Wait, actually, let's look at the vertical and horizontal segments.
First rectangle: top part, length 17 mm, width 3 mm.
Second rectangle: middle part, length \( 17 - 5 = 12 \)? No, wait, the middle indent: the horizontal length for the middle rectangle (the one with height 4 mm) is \( 17 - 5 = 12 \)? Wait, no, maybe better to calculate the area of the big rectangle minus the missing parts. Wait, the big rectangle would be 17 mm (length) and 9 mm (height), but there are two indentations. Wait, no, let's do it by adding three rectangles.

First rectangle: top, length 17 mm, width 3 mm. Area \( A_1=17\times3 = 51\) \( \text{mm}^2 \).

Second rectangle: middle, the height is 4 mm, and the length is \( 17 - 5=12 \)? Wait, no, the horizontal length: from the left, after the first 5 mm indent, the remaining length is \( 17 - 5 = 12 \), but then there is another 3 mm indent? Wait, no, let's look at the bottom part. The bottom rectangle: length 9 mm, width 2 mm. Wait, maybe:

First rectangle: top, 17mm (length) × 3mm (width): \( 17\times3 = 51 \).

Second rectangle: middle, the height is 4mm, and the length is \( 17 - 5=12 \)? No, wait, the horizontal length for the middle rectangle (the one with height 4mm) is \( 17 - 5 = 12 \)? Wait, no, the left indent is 5mm, so the length of the middle rectangle (height 4mm) is \( 17 - 5=12 \)? Then area \( A_2 = 12\times4=48 \).

Third rectangle: bottom, length 9mm, width 2mm: \( 9\times2 = 18 \).

Wait, but let's check the total height: 3 + 4+ 2 = 9, which matches the total height of 9mm. And the length: for the top rectangle, length 17mm. For the middle rectangle, length is \( 17 - 5=12 \)? Wait, no, the left side: the first indent is 5mm, so the middle rectangle's left end is at 5mm from the left, so its length is \( 17 - 5 = 12 \)? Then the bottom rectangle: length 9mm (as given on the right, the bottom length is 9mm), width 2mm.

Now sum them up: \( 51+48 + 18=117 \)? Wait, no, that can't be right. Wait, maybe a better approach: calculate the area of the large rectangle (17×9) and subtract the areas of the two missing rectangles.

Large rectangle area: \( 17\times9=153 \) \( \text{mm}^2 \).

First missing rectangle: the one with length 5mm and height \( 9 - 3=6 \)? No, wait, the first indent: height is \( 9 - 3=6 \)? No, the first indent (the upper left indent) has a height of \( 9 - 3=6 \)? Wait, no, the top rectangle is 3mm height, then the middle part: the indent is 5mm in length and \( 9 - 3=6 \) in height? No, that's not correct. Wait, the figure:

Top: 17mm (length) × 3mm (height).

Then, below the top rectangle, there is a horizontal indent of 5mm (length) and the vertical space is \( 9 - 3=6 \)mm, but within that 6mm, there is a 4mm height rectangle and a 2mm height rectangle. Wait, maybe the two missing rectangles:

First missing rectangle: length 5mm, height \( 9 - 3=6 \)mm? No, that's not right. Wait, let's look at the coordinates.

Alternative method:

The figure can be divided into three rectangles:

1. Top rectangle: length = 17 mm, width = 3 mm. Area: \( 17 \times 3 = 51 \) \( \text{mm}^2 \).

2. Middle rectangle: length = \( 17 - 5 = 12 \) mm, width = 4 mm. Area: \( 12 \times 4 = 48 \) \( \text{mm}^2 \).

3. Bottom rectangle: length = 9 mm, width = 2 mm. Area: \( 9 \times 2 = 18 \) \( \text{mm}^2 \).

Now, sum these areas: \( 51 + 48 + 18 = 117 \) \( \text{mm}^2 \).

Wait, but let's check with another method. The large rectangle (if there were no indents) would be 17 mm (length) × 9 mm (height) = 153 \( \text{mm}^2 \). Now, the two indents:

First indent: a rectangle with length 5 mm and height \( 9 - 3 = 6 \) mm? No, that's not correct. Wait, the first indent (the left - most indent) has a length of 5 mm and a height of \( 9 - 3=6 \) mm? No, the height of the indent is \( 9 - 3 = 6 \), but within that, there is a 4 mm height and a 2 mm height. Wait, no, the first indent (the upper left) is a rectangle with length 5 mm and height \( 9 - 3=6 \) mm? No, that's wrong. Wait, the correct way is to find the areas of the missing parts.

Wait, the first missing part (the upper left indent) is a rectangle with length 5 mm and height \( 9 - 3 = 6 \) mm? No, because the middle part has a height of 4 mm and the bottom part has a height of 2 mm. Wait, 3 (top) + 4 (middle) + 2 (bottom) = 9 (total height), which is correct.

The first missing rectangle (the one that is indented on the left, above the middle and bottom rectangles) has length 5 mm and height \( 4 + 2=6 \) mm. Area of this missing rectangle: \( 5\times6 = 30 \) \( \text{mm}^2 \).

The second missing rectangle (the one indented on the left of the bottom rectangle) has length \( 17 - 5 - 9=3 \) mm (because the bottom rectangle has length 9 mm, and the total length is 17 mm, so 17 - 5 - 9 = 3 mm) and height 2 mm. Area of this missing rectangle: \( 3\times2 = 6 \) \( \text{mm}^2 \).

Now, total area of large rectangle: \( 17\times9 = 153 \). Subtract the two missing areas: \( 153-30 - 6=117 \) \( \text{mm}^2 \). Which matches the previous result.

Step2: Verify the calculation

We can also calculate by adding the three rectangles:

- Top rectangle: \( 17\times3 = 51 \)
- Middle rectangle: The length is \( 17 - 5 = 12 \), height 4, so \( 12\times4 = 48 \)
- Bottom rectangle: \( 9\times2 = 18 \)

Sum: \( 51 + 48+18 = 117 \)

Answer:

117