Math Doubts

How to find the Area of a Running Track

A running track of 3.5 m wide all around which consists of two straight paths and two semi-circular rings. Find the area of the track.

The running track displays two rectangles and two semi circles. So, split it into four shapes for obtaining the total area of the running track.

Area of Two Rectangles

Let us first find the area of rectangle. We know that

$Area \, of \, Rectangle = Length \times Width$

In this question, it is given that the length of the rectangle is $140$ meters and width is $3.5$ meters.

$Area \, of \, Rectangle = 140 \times 3.5$

$\therefore Area \, of \, Rectangle = 490 \, m^2$

It is area of one rectangle. So, the area of two rectangles is twice the area of one rectangle.

$Area \, of \, 2 \, Rectangle = 2 \times 490$

$\therefore Area \, of \, 2 \, Rectangle = 980 \, m^2$

Area of Concentric circles

It is given that the diameter of the internal circle is $42$ m. So, its radius is $21$ meters.

We know that the width of the track is $3.5$ meters. Therefore, the radius of the external circle is $21 + 3.5$ meters and it is $24.5$ meters.

We know that the area of a circle is $\pi r^2$

Here, you need to use one technique, subtract area of external circle from internal circle to find the area of concentric circle.

The area of internal circle is $\pi (21)^2$

The area of external circle is $\pi(24.5)^2$

$\therefore Area \, of \, concentric \, circle = \pi(24.5)^2 -\pi(21)^2$

$\implies Area \, of \, concentric \, circle = \pi[(24.5)^2 -(21)^2]$

$\implies Area \, of \, concentric \, circle = \pi[(24.5)^2 -(21)^2]$

$\implies Area \, of \, concentric \, circle = 500.2986 \,m^2$


The area of track = Area of 2 Rectangles + Area of Concentric circle

The area of track = $980 + 500.2986$

The area of track = $1480.2986 \, m^2$

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