Divide the hexagon into six triangles by drawing all diagonals. Notice that the diagonals will bisect each other and the angles at the hexagon's vertices.
About 41.38 feet^2
Practice makes perfect
To cover the entire table, Mr. Singer needs one piece for the hexagon and six rectangular pieces for the edges.
If we find one triangle's area, we can then calculate the area of the hexagon. In order to do this we will first determine the sum of the hexagon's interior angles.
The sum of the interior angles is 720^(∘). Since this is a regular hexagon, each interior angle will measure 720^(∘)6=120^(∘). Additionally, the diagonal between two opposite vertices bisects the angles at each vertex. Therefore, if each of the hexagon's angles is 120^(∘), half of this will be 60^(∘).
In this triangle two angles are 60^(∘). This means the third angle must be 60^(∘) as well. Therefore, this is an equilateral triangle with sides of 3 feet.
By finding the height of this triangle we can determine its area and then the area of the hexagon. Notice that the height makes up a leg in a right triangle, where the hypotenuse and second leg are known. Therefore, we can find the height by the Pythagorean Theorem.
Now we can calculate the area of the triangle and then the hexagon by multiplying this number by 6.
Area hexagon: (1/2(3)sqrt(6.75))6=9sqrt(6.75) ft^2
Area of Rectangles
When determining the area of the hexagon, we found that the hexagon had a side of 3 feet, which also happens to be the length of the rectangular pieces. From the exercise, we also know that the rectangle has a width of 1 foot.
The product of the width and length of a rectangle gives us its area. If we multiply this by 6, we get the total area of the rectangular pieces.
Area rectangles: (1* 3)6=18 ft^2
Total Area
Finally, by adding the area of the hexagon and the rectangles together we can determine the total area of tablecloth that Mr. Singer needs.
Total area: 18+9sqrt(6.75)≈ 41.38ft^2
