Question

Robert is designing hexagonal tiles that are made up of 12 identical right triangles, as shown. Write three equations robert can use the find the area A of the hexagonal tile in square centimeters.

Answer 1

Question:-

Robert is designing hexagonal tiles that are made up of 12 identical right triangles, as shown. Write three equations robert can use the find the area A of the hexagonal tile in square centimeters.

Given:-

1. Hexagon is made up of identical 12 triangles
2. Sides of triangle :- 5cm,4cm,3cm

To Find:-

1. 3 Equations for finding the area of the triangle

Answer:-

First method:-

we know 12 triangles make up hexagon so we can say

12× Area of triangle = Area of hexagon

`12 * ( (1)/(2) * base * height) = area \: of \: hexagon`

`12 * (1)/(2) * 3 * 4 = area \: of \: hexagon \\ 12 * 3 * 2 = area \: of \: hexagon \\ 12 * 6 = area \: of \: hexagon \\ 72cm {}^(2) = area \: of \: hexagon`

Second method:-

we can see that there is a rectangle in the middle of the hexagon and 4 triangles (identical ) on 2 sides

According to question:-

1. sides of rectangle :- 8 cm and 6cm
2. side of triangle is same

so area of hexagon= area of square+ 4 area of triangle

`( l * b ) + (4( (1)/(2) * b * h)) = area \: of \: hexagon`

`(8 * 6 ) + 4((1)/(2) * 4 * 3) = area \: of \: hexagon \\ 48 + 24= area \: of \: hexagon \\ 72 {cm}^(2) = area \: of \: hexagon`

Third method:-

we can see 2 square at the sides of the 4 triangle at the middle so we can say

1. side of square = 5cm
2. sides of triangle = 4cm ,3cm

Area of hexagon =4× area of triangle + 2 area of square

`area \: of \: hexagon = (4(1 )/(2) × 4×3)+2(5×5)`

`area \: of \: hexagon = 24+ 50 \\ area \: of \: hexagon = 74cm² \\ area \: of \: hexagon = 72 cm ² approx`

This approximation turn out bez in your question the area of 4 identical triangle is not equal to square formed by them

4 area of triangle = area of square

4×6 = 24 cm² not equal to 5² = 25 cm²

thats why there is approximation of 2 cm²