Question

what is the area of this figure?
5 mm
7 mm
10 mm
4 mm
3 mm
2 mm
5 mm
write your answer using decimals, if necessary.
____ square millimeters
submit
work it out
not feeling ready yet? these

Explanation:

Step1: Split shape into 3 parts

Divide the figure into a rectangle, a smaller rectangle, and a triangle.

Step2: Calculate area of top rectangle

Top rectangle: length = 5 mm, height = 7 mm
Area: $5 \times 7 = 35$ mm²

Step3: Calculate area of left rectangle

Left rectangle: length = 4 mm, height = 3 mm
Area: $4 \times 3 = 12$ mm²

Step4: Calculate area of bottom triangle

First find base and height of the triangle:
Base of triangle: $5 + 4 = 9$ mm
Height of triangle: $10 - 7 - 3 = 0$? Correct: total right side =10 mm, subtract 7 mm and 3 mm, remaining vertical height for triangle is $10 - 7 = 3$ mm, horizontal base is $5 + 4 = 9$ mm? No, correct: the slanted part forms a triangle with base $5 + 4 = 9$ mm and height $10 - (7+3) + 2$? No, better: total vertical side 10 mm, the lower part's vertical length is $10 - 7 = 3$ mm, horizontal length for the lower section is $5 + 4 = 9$ mm, and the indent is 2 mm. So the bottom shape is a trapezoid? No, split correctly:
Alternative split: The figure can be a large rectangle minus a small rectangle plus a triangle? No, better:
Total shape:
1. Top rectangle: $5 \times 7 = 35$
2. Middle rectangle: $(5+4) \times 3 = 27$? No, left rectangle is 4x3=12, then the right lower part:
The right side is 10 mm, top is 7 mm, so lower right height is $10 - 7 = 3$ mm, width 5 mm, plus the triangle at the bottom: the bottom has a 2 mm indent, so the triangle has base $5 + 4 = 9$ mm, height $5 - 3 = 2$ mm? No, correct way:
Let's use coordinate system:
Top right corner (5,10), top left (0,10), then left to (0,3), right to (4,3), down to (4,0), right to (9,0), up to (5,10).
So area can be calculated via shoelace formula:
Coordinates in order: (0,10), (5,10), (5,0), (9,0), (4,0), (4,3), (0,3), (0,10)
Shoelace formula:
Area = $\frac{1}{2} |\sum_{i=1}^n (x_i y_{i+1} - x_{i+1} y_i)|$
Calculate each term:
$0*10 + 5*0 + 5*0 + 9*0 + 4*3 + 4*3 + 0*10 = 0 + 0 + 0 + 0 + 12 + 12 + 0 = 24$
$10*5 + 10*5 + 0*9 + 0*4 + 0*4 + 3*0 + 3*0 = 50 + 50 + 0 + 0 + 0 + 0 + 0 = 100$
Difference: $|24 - 100| = 76$
Area = $\frac{1}{2} * 76 = 38$? No, wrong coordinates. Correct coordinates:
Top left (0,10), top right (5,10), down to (5,0), then left to (0,0) is not the shape. Correct shape:
Top: (0,10) to (5,10) (5 mm)
Down to (5,3) (7 mm down, 10-7=3)
Left to (5-4,3) = (1,3) (4 mm left)
Down to (1,0) (3 mm down, 3-0=3? No, the left side has 3 mm, then 2 mm indent, then 5 mm down. So (1,3) down to (1,3-3)= (1,0)? No, the 2 mm is indent, so (1,3) right to (1+2,3)=(3,3), down to (3,3-5)=(3,-2)? No, this is confusing.
Alternative split:
1. Top rectangle: 5 mm x 7 mm = 35 mm²
2. Middle rectangle: 4 mm x 3 mm = 12 mm²
3. Bottom trapezoid: two parallel sides: 5 mm and (5+4) mm, height = (10 -7 -3) +5? No, 10-7-3=0, so the bottom is a triangle with base (5+4)=9 mm, height 2 mm? No, the 5 mm is the vertical side, so area of triangle is $\frac{1}{2} * 9 * 2 = 9$ mm²? No, total area would be 35+12+9=56? No, wrong.
Correct split:
The figure can be viewed as a rectangle of 10 mm x 9 mm (total width 5+4=9, total height 10) minus a rectangle of 7 mm x4 mm (the top left indent) minus a triangle of 4 mm x 2 mm? No, 10*9=90, minus 7*4=28, minus $\frac{1}{2}*4*2=4$, 90-28-4=58? No.
Wait, let's calculate the missing parts:
Total bounding box: width 5+4=9 mm, height 10 mm. Area=9*10=90 mm².
Subtract the top left indent: a rectangle of 4 mm x (10-7)=3 mm? No, top left indent is 4 mm wide, 10-7=3 mm tall? No, the top left has a 7 mm tall rectangle, so the indent is 4 mm x (10-7)=3 mm? No, the left side has 3 mm, then 2 mm indent, then 5 mm down.
The indent at the bottom left: a rectangle of 4 mm x 2 mm, and a triangle of 4 mm x (10-7-3)=0? No, 10-7-3=0, so the bottom indent is a triangle of 4 mm x 2 mm? No, 2 mm is the horizontal indent, 5 mm is the vertical side, so the missing triangle has base 4 mm, height 2 mm, area $\frac{1}{2}*4*2=4$ mm².
The top left missing rectangle is 4 mm x (10-7)=3 mm, area 12 mm².
So total area = 90 -12 -4=74? No, this is wrong.
Alternative correct way:
Use the shoelace formula with correct coordinates:
List all corners in order (clockwise):
1. (0, 10) - top left of top rectangle
2. (5, 10) - top right of top rectangle
3. (5, 0) - bottom right of the figure
4. (0, 5) - bottom left of the slanted side (since 10-5=5, and the indent is 2 mm, so (0+4,5)=(4,5)? No, the left side: (0,10) down to (0,7) (7 mm), then right to (4,7) (4 mm), down to (4,4) (3 mm), right to (6,4) (2 mm), down to (6,0) (4 mm), left to (5,0) (1 mm)? No, the labels are:
Top side: 5 mm
Left top side:7 mm
Left middle:4 mm, 3 mm
Left bottom:2 mm,5 mm
Right side:10 mm
So correct coordinates:
- Top right: (5, 10)
- Top left: (0, 10) (5 mm left from top right)
- Down 7 mm to (0, 3)
- Right 4 mm to (4, 3)
- Down 3 mm to (4, 0)
- Right 2 mm to (6, 0)
- Up 5 mm to (6, 5)
- Right to (5, 10)? No, right side is 10 mm, so (5,10) to (5,0) is 10 mm, then (5,0) to (9,0) is 4 mm, up to (9,5) is 5 mm, left to (5,10) is the slant. Oh! Yes! That's the correct shape:
1. (5, 10) top right
2. (0, 10) top left (5 mm left)
3. (0, 3) down 7 mm
4. (4, 3) right 4 mm
5. (4, 0) down 3 mm
6. (9, 0) right 5 mm (5+4=9)
7. (5, 10) up to top right (slant side)
Now apply shoelace formula:
$x_1=5,y_1=10; x_2=0,y_2=10; x_3=0,y_3=3; x_4=4,y_4=3; x_5=4,y_5=0; x_6=9,y_6=0; x_7=5,y_7=10$
Calculate sum of $x_i y_{i+1}$:
$5*10 + 0*3 + 0*3 + 4*0 + 4*0 + 9*10 + 5*10$
$=50 + 0 + 0 + 0 + 0 + 90 + 50 = 190$
Calculate sum of $y_i x_{i+1}$:
$10*0 + 10*0 + 3*4 + 3*4 + 0*9 + 0*5 + 10*5$
$=0 + 0 + 12 + 12 + 0 + 0 + 50 = 74$
Area = $\frac{1}{2} |190 - 74| = \frac{1}{2} * 116 = 58$ mm²

Step5: Sum all areas

Top rectangle (5x7=35) + left rectangle (4x3=12) + trapezoid (area of lower part: $\frac{(5+9)*3}{2} + \frac{9*2}{2}$? No, shoelace gives 58, which is correct.

Answer:

58 square millimeters