Area Calculation of a Park Plot

MathematicsGeometryMedium

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Question

Calculate the total area of the green shaded region. The green region is defined by the outer polygon with the following total dimensions: top width $46\text{ m}$, total left height $50\text{ m}$, bottom width $60\text{ m}$, and right vertical edge $22\text{ m}$. Within this area, subtract the following: the circular cutout with diameter $14\text{ m}$ in the top left, the $5\text{ m} \times 5\text{ m}$ square fountain, the L-shaped playground, and the flower garden.

This question includes visual content: The image shows a large, complex, polygon-shaped plot of land representing a park, with several white-colored cutout sections inside it. The overall boundary of the park is defined by given dimensions: a total width of 60 m at the bottom, a total height of 50 m on the left, an upper edge of 46 m, and a right-side vertical segment of 22 m. Inside the green shaded area, there are four features: 1) A circular cutout at the top-left corner with a diameter of 14 m. 2) A square 'Fountain' with dimensions 5 m by 5 m. 3) A complex-shaped 'Playground' (an L-shape comprised of a 5x20 m rectangle and a larger rectangle). 4) A 'Flower garden' represented by a shape with a rectangular center of 7x15 m and rounded or angled edges, with dimensions labeled 11.5 m and 7.94 m.

Animated Video Solution

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Step by Step Written Solution

1
Step 1

Hi Keita, let's work through this composite area problem step by step and find the shaded garden area.

Find the Shaded Area

We will calculate the total outer area and subtract the areas of the cut-out sections.

2
Step 2

First, notice that the outer boundary can be split into a large rectangle and a triangle at the top right.

$$A_{rectangle}=60\times50$$
$$A_{triangle}=\frac12\times(60-46)\times(50-22)$$
3
Step 3

The rectangle area is three thousand square metres.

4
Step 4

The triangle has base fourteen metres and height twenty eight metres, giving an area of one hundred ninety six square metres.

5
Step 5

So the total outer area is three thousand minus one hundred ninety six, which equals two thousand eight hundred four square metres.

$$A_{outer}=3000-196=2804\text{ m}^2$$
6
Step 6

Now let's calculate the areas that are not part of the shaded region.

Areas to Subtract

$$A_{semicircle}=\frac12\pi r^2$$
$$A_{playground}=28\times20-(28-5)\times7$$
$$A_{fountain}=5\times5$$
7
Step 7

The semicircle has diameter fourteen metres, so the radius is seven metres.

8
Step 8

Using pi equal to twenty two over seven, the semicircle area becomes seventy seven square metres.

9
Step 9

For the playground, start with the large rectangle and subtract the missing top corner.

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About This Question

Subject
Mathematics
Topic
Geometry
Difficulty
Medium
Question Type
Open Ended

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