A triangle is inside a circle where the triangle’s base is on the circle’s diameter as shown. What is the area of the shaded region? Use 3.14 for
. Enter your answer as a decimal in the box. ft
16 ft
The Correct Answer and Explanation is:
Step 1: Understand the given dimensions
- The diameter of the circle is 16 ft.
- Since the triangle’s base is on the circle’s diameter and the triangle is inscribed, it forms a right triangle (by the Thales’ Theorem).
- The base of the triangle = diameter = 16 ft
- Since it is an isosceles right triangle (as shown), the two equal legs are from the ends of the diameter up to the top of the circle, forming the height.
To find the height of the triangle, note that the radius = half of diameter = 8 ft, and the triangle is right-angled with the right angle at the top. So the height = radius = 8 ft
Step 2: Area of the Circle
Areacircle=πr2=3.14×(8)2=3.14×64=200.96 ft2\text{Area}_{\text{circle}} = \pi r^2 = 3.14 \times (8)^2 = 3.14 \times 64 = 200.96 \text{ ft}^2Areacircle=πr2=3.14×(8)2=3.14×64=200.96 ft2
Step 3: Area of the Triangle
Areatriangle=12×base×height=12×16×8=64 ft2\text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 16 \times 8 = 64 \text{ ft}^2Areatriangle=21×base×height=21×16×8=64 ft2
Step 4: Area of the Shaded Region
Shaded Area=Areacircle−Areatriangle=200.96−64=136.96 ft2\text{Shaded Area} = \text{Area}_{\text{circle}} – \text{Area}_{\text{triangle}} = 200.96 – 64 = \boxed{136.96 \text{ ft}^2}Shaded Area=Areacircle−Areatriangle=200.96−64=136.96 ft2
Explanation
In this problem, a triangle is inscribed in a circle with its base along the diameter. This is a classic geometric configuration where the triangle is a right triangle because of Thales’ Theorem, which states that any triangle inscribed in a circle with the diameter as one side is a right triangle. The right angle is opposite the diameter, meaning the triangle’s height is the radius of the circle.
We are asked to find the area of the shaded region, which is the part of the circle not occupied by the triangle. This involves two key geometric formulas: the area of a circle and the area of a triangle. The radius of the circle is half the diameter, so here it is 8 ft. Using the formula for the area of a circle A=πr2A = \pi r^2A=πr2, and substituting r=8r = 8r=8 and π=3.14\pi = 3.14π=3.14, the total circle area is 200.96 ft².
The triangle, being a right triangle with base 16 ft and height 8 ft, has an area of A=12×base×height=64A = \frac{1}{2} \times \text{base} \times \text{height} = 64A=21×base×height=64 ft².
To get the shaded area, we subtract the triangle’s area from the circle’s area: 200.96−64=136.96 ft2200.96 – 64 = 136.96 \text{ ft}^2200.96−64=136.96 ft2
This result represents the area outside the triangle but still within the circle. This problem demonstrates the importance of understanding basic geometric properties and relationships in composite shapes.
✅ Final Answer: 136.96 ft²
