Question

what is the area of the shaded region?
40 in
26 in
18 in
27 in
write your answer as a whole number or
a decimal rounded to the nearest
hundredth.
square inches

Explanation:

Step1: Calculate area of large triangle

The formula for the area of a triangle is $A = \frac{1}{2} \times base \times height$. For the large triangle, base = 27 in, height = 40 in.
$A_{large} = \frac{1}{2} \times 27 \times 40 = 540$ square inches.

Step2: Calculate area of small triangle

For the small triangle, base = 18 in, height: we can check if it's a right triangle (since 18-26-? might be, but using Pythagoras: $height = \sqrt{26^2 - 18^2} = \sqrt{676 - 324} = \sqrt{352} \approx 18.76$? Wait, no, wait the large triangle is right-angled? Wait, the large triangle has legs 40 and 27? Wait, no, looking at the diagram, the large triangle is a right triangle with legs 40 in (vertical) and 27 in (horizontal). The small triangle inside is also a right triangle with base 18 in and hypotenuse 26 in? Wait, no, maybe the small triangle is right-angled with base 18 and height such that hypotenuse is 26. Wait, but actually, maybe the large triangle is right-angled, so area is 1/2 * 40 *27. The small triangle: let's check if it's right-angled. 18-24-30? No, 18 and 24: 18²+24²= 324+576=900=30², but here hypotenuse is 26. Wait, maybe I made a mistake. Wait, the small triangle: base 18, and the other leg: let's calculate. $h = \sqrt{26^2 - 18^2} = \sqrt{676 - 324} = \sqrt{352} \approx 18.76$? No, that can't be. Wait, maybe the large triangle is right-angled, so area is 1/2 * 40 *27 = 540. The small triangle: maybe it's a right triangle with base 18 and height 24? Wait, 18-24-30, but hypotenuse here is 26. Wait, maybe the diagram is a right triangle, large with legs 40 and 27, small with legs 18 and (let's see, maybe the small triangle's height is such that it's similar? Wait, no, the problem is to find the area of the shaded region, which is large triangle area minus small triangle area.

Wait, maybe the small triangle is a right triangle with base 18 and height calculated as follows: Wait, 18 and 24: 18² +24²= 324 + 576= 900=30², but here hypotenuse is 26. Wait, maybe I misread the diagram. Wait, the large triangle: vertical side 40, horizontal side 27, so it's a right triangle. The small triangle: base 18, and the other leg: let's check the hypotenuse 26. So $height = \sqrt{26^2 - 18^2} = \sqrt{676 - 324} = \sqrt{352} \approx 18.76$? No, that's not an integer. Wait, maybe the small triangle is a right triangle with legs 18 and 24? Wait, 18*24/2=216. Then large triangle area is 40*27/2=540. Then shaded area is 540 - 216=324? But wait, 18-24-30, but the hypotenuse here is 26. Wait, maybe the diagram has a typo, but maybe the small triangle is right-angled with base 18 and height 24 (since 18-24-30, but maybe the hypotenuse is 30, but here it's 26. Wait, no, maybe I made a mistake. Wait, let's recalculate.

Wait, the large triangle: base 27, height 40, area = 0.5*27*40 = 540.

Small triangle: base 18, let's find its height. If the small triangle is right-angled, then hypotenuse 26, so height = sqrt(26² - 18²) = sqrt(676 - 324) = sqrt(352) ≈ 18.76166. Then area of small triangle is 0.5*18*18.76166 ≈ 0.5*18*18.76166 ≈ 168.8549. Then shaded area is 540 - 168.8549 ≈ 371.1451, which is about 371.15. But that seems odd. Wait, maybe the small triangle is a right triangle with legs 18 and 24 (since 18-24-30), but the hypotenuse is 30, not 26. Maybe the diagram has a mistake, but maybe the 26 is a typo, and it's 30. Then small triangle area is 0.5*18*24=216, shaded area 540-216=324. But the problem says 26. Wait, let's check again.

Wait, 18² + 24² = 324 + 576 = 900 = 30². So 18-24-30. So if the hypotenuse is 26, then the other leg is sqrt(26² - 18²)=sqrt(676-324)=sqrt(352)=sqrt(16*22)=4*sqrt(22)≈4*4.690≈18.76. So area of small triangle is 0.5*18*18.76≈168.84. Then large triangle area is 0.5*27*40=540. So shaded area is 540 - 168.84=371.16. Wait, maybe that's the answer.

Wait, let's do the calculation properly.

Large triangle area:
$A_{large} = \frac{1}{2} \times 27 \times 40 = 540$

Small triangle: base = 18, hypotenuse = 26, so height (let's call it h) is:
$h = \sqrt{26^2 - 18^2} = \sqrt{676 - 324} = \sqrt{352} = \sqrt{16 \times 22} = 4\sqrt{22} \approx 4 \times 4.6905 = 18.762$

Area of small triangle:
$A_{small} = \frac{1}{2} \times 18 \times 18.762 \approx \frac{1}{2} \times 337.716 = 168.858$

Shaded area:
$A_{shaded} = A_{large} - A_{small} \approx 540 - 168.858 = 371.142$

Rounded to the nearest hundredth, that's 371.14. Wait, but maybe the small triangle is a right triangle with legs 18 and 24 (since 18-24-30), but the hypotenuse is 30, not 26. Maybe the diagram has a mistake, but according to the given numbers, we have to use 26. So the calculation is as above.

Wait, maybe I made a mistake in assuming the large triangle is right-angled. Wait, the diagram shows a large triangle with vertical side 40, horizontal side 27, so it's a right triangle. The small triangle inside is also a right triangle with base 18 and hypotenuse 26. So the height of the small triangle is sqrt(26² - 18²)≈18.76. Then area of small triangle is 0.5*18*18.76≈168.84. Then shaded area is 540 - 168.84=371.16. Wait, let's compute sqrt(352) more accurately. 352=16*22, so sqrt(352)=4*sqrt(22)≈4*4.69041576≈18.76166304. Then 0.5*18*18.76166304=9*18.76166304≈168.8549674. Then 540 - 168.8549674≈371.1450326, which is approximately 371.15 when rounded to the nearest hundredth.

Answer:

371.15