A playground consists of a rectangular field that measures 10 meters by 8 meters. A semicircular sandbox is constructed along one of the -meter sides of the rectangle so that the diameter of the semicircle lies on that side. What is the total area, in square meters, of the playground including the sandbox?

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SAT Area & Volume Question 35 | Free SAT Prep | Aniko

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Question 35·Medium·Area and Volume

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A playground consists of a rectangular field that measures 10 meters by 8 meters. A semicircular sandbox is constructed along one of the $8$ -meter sides of the rectangle so that the diameter of the semicircle lies on that side.

What is the total area, in square meters, of the playground including the sandbox?

For composite-area problems on the SAT, mentally split the figure into basic shapes (rectangles, circles, triangles), find each area using the appropriate formula, and then add or subtract as the diagram indicates. Pay close attention to relationships like “diameter lies on this side” so you can get the radius correctly, and remember that semicircles and quarter circles are fixed fractions of a full circle’s area: start with $\pi r^2$ and then multiply by $\tfrac12$ or $\tfrac14$ as needed. Keep your result in terms of $\pi$ so it’s easy to match to the multiple-choice options.

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Identify all parts of the shape

Think of the playground as a rectangle with a semicircle attached to one of its sides. How can you express the total area using the areas of these simpler shapes?

Work out the semicircle’s radius

The semicircle’s diameter lies along an $8$ -meter side of the rectangle. If the diameter is $8$ meters, what is the radius?

Use the correct area formula and fraction of a circle

Use the circle area formula $A = \pi r^2$ with your radius. Then adjust your result because you only have a semicircle, not a full circle, and finally add this to the rectangle’s area.

Desmos Guide

Compute the total area expression

In Desmos, enter the expression 10*8 + 0.5*pi*4^2 to represent the rectangle’s area plus the semicircle’s area. Look at the simplified symbolic result (in terms of $\pi$ ) and match that expression to one of the answer choices.

Step-by-step Explanation

Break the shape into simpler parts

The playground is made of:

A rectangle measuring $10$ m by $8$ m
A semicircle attached along one of the $8$ -meter sides

The total area is the sum of the rectangle’s area and the semicircle’s area.

First, find the rectangle’s area:

Area of a rectangle = (length) $\times$ (width)
So the rectangle’s area is $10\times 8 = 80$ square meters.

Find the radius of the semicircle

The diameter of the semicircle lies along the $8$ -meter side of the rectangle, so the diameter is $8$ meters.

The radius is half of the diameter:

Radius $r = 8\div 2 = 4$ meters.

Compute the area of the semicircle

Start with the area formula for a full circle: $A = \pi r^2$ .

For radius $r = 4$ :

\text{Area of full circle} = \pi\cdot 4^2 = 16\pi

A semicircle is half of a full circle:

\text{Area of semicircle} = \frac12\cdot 16\pi = 8\pi

Add the areas to get the total

Now add the rectangle’s area and the semicircle’s area:

Rectangle: $80$
Semicircle: $8\pi$

Total area:

80 + 8\pi

So the total area of the playground, in square meters, is $80+8\pi$ , which corresponds to choice C.

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation