Area Calculation: 4×8 Rectangle with Four Semicircles

Q: How do I find the radius of each semicircle?

The radius of each semicircle equals half the length of the side it's attached to. So for an 8 cm side: radius = 8 ÷ 2 = 4 cm.

Q: Do I add all four semicircle areas?

Yes! Since there are four separate semicircles, calculate each area using $ \frac{1}{2}\pi r^2 $ and add them all to the rectangle area.

Q: Why isn't the answer just the rectangle plus a full circle?

Because we have four separate semicircles of different sizes, not one circle! Two have radius 4 cm and two have radius 2 cm, so we can't combine them into one circle.

Q: How can I double-check my semicircle calculations?

Use the pattern: opposite semicircles are identical. The two on 8 cm sides both give 8π, and the two on 4 cm sides both give 2π. Total semicircle area = 2(8π) + 2(2π) = 20π.

Composite Area with Semicircle Additions

Observe the rectangle in the figure below.

A semicircle has been added to each side of the rectangle.

Determine the area of the entire shape?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's find the total area of the shape.

00:15 The total area is the rectangle's area plus the areas of four half-circles.

00:28 Each pair of half-circles are equal since they're on equal sides.

00:36 So, here is the correct formula.

00:40 Our formula involves half of the circle's area formula.

00:45 The rectangle's side, which is the diameter, measures eight.

00:51 Let's use the circle's area formula with the radius being four.

00:58 This is the area of the first half-circle, and the second is the same.

01:03 Now, we'll calculate the area of half-circle three in the same way.

01:08 Here, the diameter of the circle is four.

01:11 Let's input the correct values to find the area of half-circle three.

01:16 This is the area of half-circle three, and four is the same.

01:20 Next, calculate the rectangle area. Four times eight.

01:25 Finally, let's add up all the areas to get the total area of the shape.

01:31 And that's how we solve the problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Observe the rectangle in the figure below.

A semicircle has been added to each side of the rectangle.

Determine the area of the entire shape?

Step-by-step solution

The area of the entire shape equals the area of the rectangle plus the area of each of the semicircles.

Let's begin by labelling each semicircle with a number:

Therefore, we can determine that:

The area of the entire shape equals the area of the rectangle plus 2A1+2A3

Let's proceed to calculate the area of semicircle A1:

$\frac{1}{2}\pi r^2$

$\frac{1}{2}\pi4^2=8\pi$

Let's now calculate the area of semicircle A3:

$\frac{1}{2}\pi r^2$

$\frac{1}{2}\pi2^2=2\pi$

Therefore the area of the rectangle equals:

$4\times8=32$

Finally we can calculate the total area of the shape:

$32+2\times8\pi+2\times2\pi=32+16\pi+4\pi=32+20\pi$

Final Answer

$32+20\pi$ cm².

Key Points to Remember

Essential concepts to master this topic

Rectangle Area: Length × width = 4 × 8 = 32 cm²
Semicircle Area: Use $\frac{1}{2}\pi r^2$ where radius = half the side length
Check: Total = 32 + 8π + 8π + 2π + 2π = 32 + 20π ✓

Common Mistakes

Avoid these frequent errors

Using diameter instead of radius for semicircle area
Don't use the full side length as radius = wrong area calculation! For side length 8, radius is 4, not 8. Using 8 gives 32π instead of 8π. Always use radius = side length ÷ 2 for semicircles attached to rectangle sides.

Practice Quiz

Test your knowledge with interactive questions

Look at the rectangle below.

Side DC has a length of 1.5 cm and side AD has a length of 9.5 cm.

What is the perimeter of the rectangle?

FAQ

Everything you need to know about this question

Why are the semicircle radii different sizes?

The semicircles are attached to different sides of the rectangle! The semicircles on the 8 cm sides have radius = 4 cm, while semicircles on the 4 cm sides have radius = 2 cm.

How do I find the radius of each semicircle?

The radius of each semicircle equals half the length of the side it's attached to. So for an 8 cm side: radius = 8 ÷ 2 = 4 cm.

Do I add all four semicircle areas?

Yes! Since there are four separate semicircles, calculate each area using $\frac{1}{2}\pi r^2$ and add them all to the rectangle area.

Why isn't the answer just the rectangle plus a full circle?

Because we have four separate semicircles of different sizes, not one circle! Two have radius 4 cm and two have radius 2 cm, so we can't combine them into one circle.

How can I double-check my semicircle calculations?

Use the pattern: opposite semicircles are identical. The two on 8 cm sides both give 8π, and the two on 4 cm sides both give 2π. Total semicircle area = 2(8π) + 2(2π) = 20π.

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Area Calculation: 4×8 Rectangle with Four Semicircles

Composite Area with Semicircle Additions

❤️ Continue Your Math Journey!

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 are the semicircle radii different sizes?

How do I find the radius of each semicircle?

Do I add all four semicircle areas?

Why isn't the answer just the rectangle plus a full circle?

How can I double-check my semicircle calculations?

🌟 Unlock Your Math Potential

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