The design on a tile is made of isosceles triangles.

The side lengths of the triangles are $6\mathrm{cm}, 6\mathrm{cm} \mathrm{and} 8\mathrm{cm}.$

How much area of the tile is black?

In one of the residential colonies of Bangalore, there is a triangular piece of land (No Mans Land) that is bounded by three roads. The length of the road segments that determine the sides of the triangular land are $5 \mathrm{m}, 5 \mathrm{m}$ and $8 \mathrm{m}$.

The corporation widens the roads and the “No Man’s Land" takes the shape of an equilateral triangle of sides $5 \mathrm{m}$ each.

How much area of the “No man’s land" did the corporation use to widen its road?

The Bermuda Triangle, also known as the Devil's Triangle, is an urban legend focused on a loosely-defined region in the western part of the North Atlantic Ocean where a number of aircraft and ships are said to have disappeared under mysterious circumstances.

The figure below represents the Bermuda Triangle. Its sides are approximately$1400 \mathrm{km}, 1500 \mathrm{km}$ and $2100 \mathrm{km}$. Find its area using Heron's formula

The perimeter of the top surface of the given stool is $84$ $\mathrm{inches}$. Two of its sides are $26$and $28$$\mathrm{inches}$. The top surface of the stool is covered with cloth up to a depth of $3$$\mathrm{inches}$(observe the extended sides of the cloth) as shown below.

Sunil needs to buy a cloth piece to stitch a cover for the stool. Its width is $3$$\mathrm{ft}$ft.

What is the shortest length of the cloth that Sunil needs to buy?

(Note: Assume $1 \mathrm{inch} = 2.54 \mathrm{cm}$ and $1\mathrm{ft} = 12 \mathrm{inches}$. Round your answer to the nearest whole number. )

Umesh went to the Calicut beach with his parents on Sunday. He saw a beautiful triangular-shaped stone. He took the stone home and the next day he brought that stone to school to show his friends.  They observed that the perimeter of the stone is $40 \mathrm{cm}$ and its two sides are $15 \mathrm{cm}$ and $11 \mathrm{cm}$.

On that day, their Mathematics teacher taught them Heron's formula to find the area of the triangle. So, Umesh and two of his friends decided to determine the area of the stone using the Herons formula and their answer are:

Mini: Area $=\frac{1}{2}\times 15\times 11 \mathrm{sq}.\mathrm{cm}$

Seema: Area $=\frac{1}{2}\times 15\times 14 \mathrm{sq} \mathrm{cm}$

Umesh: Area $=\sqrt{20\times 11\times 15} \mathrm{sq} \mathrm{cm}$

Who answered it correctly.

Mahesh has a livestock of $2$ cows and $3$ goats. He wants to prepare a triangular fenced field where they can graze safely. For fencing the land, Mahesh went to the market and bought a rope of $24 \mathrm{m}$ length.

The largest triangular area which Mahesh will be able to form with rope will be