Calculate Trapezoid Area with 7π Segment and Semicircle: Step-by-Step Solution

Q: Why is the arc length 7π and not the full circumference?

The diagram shows a semicircle, which is only half of a full circle. The arc length of a semicircle is $ \pi r $, while a full circle would be $ 2\pi r $.

Q: How do I know which measurement is the upper base?

The upper base is the straight line segment that the semicircle sits on. This equals the diameter of the semicircle, which is twice the radius.

Q: What if I confused the height with another measurement?

Look for the perpendicular distance between the parallel bases. In this problem, it's labeled as 7 and shown as a vertical red line from top to bottom.

Q: Why does the semicircle affect the trapezoid area calculation?

The semicircle doesn't change the area formula, but it defines the length of the upper base. The semicircle's diameter becomes the trapezoid's upper base measurement.

Q: How do I check if 112 is the right answer?

Verify each step: $ 7\pi = \pi r $ gives r = 7, diameter = 14, then $ \frac{1}{2}(14 + 18) \times 7 = \frac{1}{2} \times 32 \times 7 = 112 $ ✓

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the area of the trapezoid

00:04 Let's draw the continuation of the circle

00:08 The arc is half the circumference of the circle according to the given data

00:12 The formula for calculating circle circumference

00:18 Let's substitute appropriate values in the formula to find the circle's radius

00:26 This is the circle's radius

00:31 Side AB is a diameter in the circle

00:35 This is the length of side AB

00:42 Let's use the formula for calculating trapezoid area

00:46 (Sum of bases(AB+DC) multiplied by height (AE)) divided by 2

00:54 Let's substitute appropriate values and calculate to find the area

01:05 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

In the drawing, a trapezoid is given, with a semicircle at its upper base.

The length of the highlighted segment in cm is $7\pi$

Calculate the area of the trapezoid

2

Step-by-step solution

To solve the problem of finding the area of the trapezoid with a semicircle on its top base, we follow these steps:

Step 1: Identify the semicircle's radius from the given circumference.
Step 2: Find the upper base length, which is the diameter of the semicircle.
Step 3: Calculate the trapezoid's area using the area formula.

Let's work through each step:

Step 1: The given length of the highlighted segment is $7\pi$ , which is the half-circumference of a circle (since it's a semicircle). The formula for the circumference of a full circle is $2\pi r$ , so for a semicircle, it is $\pi r$ . Setting this equal to the length given:

\pi r = 7\pi

Canceling $\pi$ from both sides, we find:

r = 7

Step 2: The diameter of the semicircle is twice the radius, hence:

\text{Diameter} = 2 \times 7 = 14 \, \text{cm}

This diameter also serves as the length of the upper base of the trapezoid.

Step 3: We use the formula for the area of a trapezoid:

\text{Area} = \frac{1}{2} \times (\text{Base}_1 + \text{Base}_2) \times \text{Height}

Substitute the known values ( $\text{Base}_1 = 14$ , $\text{Base}_2 = 18$ , $\text{Height} = 7$ ):

\text{Area} = \frac{1}{2} \times (14 + 18) \times 7

\text{Area} = \frac{1}{2} \times 32 \times 7

\text{Area} = 16 \times 7

\text{Area} = 112 \, \text{cm}^2

Thus, the area of the trapezoid is 112 cm $^2$ .

3

Final Answer

112

Key Points to Remember

Essential concepts to master this topic

Arc Length: Semicircle arc equals $\pi r$ , not full circumference
Technique: From $7\pi = \pi r$ , divide by π to find r = 7
Check: Verify diameter = 14, then Area = ½(14 + 18) × 7 = 112 ✓

Common Mistakes

Avoid these frequent errors

Using full circumference formula for semicircle
Don't use $2\pi r = 7\pi$ which gives r = 3.5 and wrong area = 84! A semicircle arc is only half the full circumference. Always use $\pi r = 7\pi$ for semicircle arc length.

Practice Quiz

Test your knowledge with interactive questions

$$ r=2 $$

Calculate the circumference.

FAQ

Everything you need to know about this question

Why is the arc length 7π and not the full circumference?

+

The diagram shows a semicircle, which is only half of a full circle. The arc length of a semicircle is $\pi r$ , while a full circle would be $2\pi r$ .

How do I know which measurement is the upper base?

+

The upper base is the straight line segment that the semicircle sits on. This equals the diameter of the semicircle, which is twice the radius.

What if I confused the height with another measurement?

+

Look for the perpendicular distance between the parallel bases. In this problem, it's labeled as 7 and shown as a vertical red line from top to bottom.

Can I solve this without finding the radius first?

+

No - you need the radius to find the diameter, which gives you the upper base length. The sequence is: arc length → radius → diameter → upper base → trapezoid area.

Why does the semicircle affect the trapezoid area calculation?

+

The semicircle doesn't change the area formula, but it defines the length of the upper base. The semicircle's diameter becomes the trapezoid's upper base measurement.

How do I check if 112 is the right answer?

+

Verify each step: $7\pi = \pi r$ gives r = 7, diameter = 14, then $\frac{1}{2}(14 + 18) \times 7 = \frac{1}{2} \times 32 \times 7 = 112$ ✓

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Calculate Trapezoid Area with 7π Segment and Semicircle: Step-by-Step Solution

Trapezoid Area with Semicircle Arc Length

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Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why is the arc length 7π and not the full circumference?

How do I know which measurement is the upper base?

What if I confused the height with another measurement?

Can I solve this without finding the radius first?

Why does the semicircle affect the trapezoid area calculation?

How do I check if 112 is the right answer?

🌟 Unlock Your Math Potential

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