Answer:
Piece A- 25%
Piece B- 6.25%
Piece C- 3.125%
Piece D- 15.625%
Piece E- 37.5%
Piece F- 12.5%
Explanation:
To find each shape's area as a percentage, each shape should be compared to the whole larger square, so we will first find the total area in the large square.
The question states that the square is 4cm x 4 cm, and using the equation for the area of a square, length x width, we know that the large square's area is 16cm squared.
Square A is 2cm long and 2cm wide, so similarly, we can use the length x width formula to get 4cm squared for its area. To find its percentage relative to the larger square, we must divide its area, 4, by the large square's area, 16. 4/16 = 0.25, which can be converted to a percentage by multiplying it by 100%. This means that piece A takes up 25% of the large square's area.
Square B is 1cm long and 1cm wide. Once again, we will multiply its length and width to get an area of 1cm squared. Again, we must divide this number by the total area of the large square. 1/16= 0.0625, or, after converting it to a percentage, 6.25%.
Piece C is triangular, and so we will use the formula for an area of a triangle instead: Base x height x 1/2. We know from seeing it above square B that its base is 1cm wide. It also ends vertically at square A. Because it starts after square B, which is 1cm long, and square A is 2cm long, we know that the triangle's height is 2cm - 1cm, or, 1cm long. The base multiplied by the height is 1, and dividing it by 2 gets us 1/2cm squared for its area. To find the percentage, we divide 1/2 by 16, which gives us 0.03125, or 3.125%.
Piece D is an irregular shape, however we can instead imagine it as two separate shapes- a square near the bottom and a triangle on top. The square section is 1cm wide, since the large square's side is 4cm and so far, squares A and B are lying on that side, taking up 3cm. it is also as tall as square B, making it 1cm long vertically. Using the formula for the area of a square, length x width, we know that the area of that section is 1cm squared.
The triangle section is right on top of it, making its base 1cm wide as well. The side of the triangle goes until the top of the large square, and because the square section already took up 1cm of the 4cm long side, we know that the triangle is 3cm tall. Using the area of a triangle formula, base x height x 1/2, we can multiply 1 x 3 x 1/2 to get an area of 3/2cm squared.
To find the total area of piece D, we then have to add the area of the square and triangle sections together. 1 + 3/2 = 5/2cm squared area total. Dividing 5/2 by 16 gets 0.15625, or 15.625% of the large square's area.
Piece E as well can be split up into a square and a triangle. The square section ends where square A ends, and takes up half of the large square's side, meaning it is 2cm wide and 2 cm long. Multiplying its length and width gets an area of 4cm squared.
The triangle also takes up half of the larger square's side horizontally and vertically, so it has a base and height of 2cm. Using the area of a triangle formula, base x height x 1/2, we can get 2 x 2 x 1/2, or 2cm squared for its area.
Adding the areas of the square and triangle sections, Piece E will have a total area of 6cm squared. Then, we divide that by 16, and 6/16= 0.375, or 37.5% of the large square's area.
Piece F is diagonal, so it is harder to tell the measurements of its base and height. however, we know that its sides are the hypotenuses of two other triangles we have measured. The hypotenuse is the longest side of a triangle, facing opposite of its 90 degree angle. If we name the hypotenuse "c" and the other two sides of the triangles "a" and "b", we can use the Pythagorean theorem, a^2+b^2=c^2, to find the lengths of Piece F.
For piece C, the base and height of the triangle are both 1cm, so if those are a and b, then 1^2+1^2=c^2, with c being the base of piece F. 1*1=1, so 1+1=c^2, making c^2=2. We can then isolate c by taking the square root of both sides, making c=
.
We can do the same thing with the triangle taken from piece E, though this time a and b will both equal 2cm. If we plug those values into the equation, then 2^2+2^2=c^2. 2*2=4, so 4+4=c^2, or c^2=8. Taking the square root of both sides, we find that c=
.
Basically, piece F has a base of
cm, and a height of
cm. We can plug those into the equation for the area of a triangle, base x height x 1/2, to get
x
x 1/2, or 2cm squared for the area of piece F. 2/16= 0.125, so piece F takes up 12.5% of the large square's area.