Shapes inside shapes

Here is a rectangle. It is 40 centimetres long and 14 centimetres wide. Two squares inside it, one at each end. Now a quarter circle at every corner of each square. The part outside those 8 quarter circles is shaded. Find the area of the shaded part. Take pi as 22 over 7. Watch how one is drawn. Put the point of the compasses on this corner, and swing the pencil from one side of the square round to the other. That is one quarter circle. Now the same at the next corner. And the next. And the next. Four corners, so 4 quarter circles, all the same size. And the other square the same way. 8 quarter circles altogether. The part outside those 8 quarter circles is shaded. Blue is the shaded part. So that blue part is the area to find. And pi is 22 over 7. Before any pi, read the lengths down through the shapes. The rectangle is 14 wide. That same 14 is the side of a square. And on that side, 2 quarter circles meet at the middle. So the side is a whole diameter. The pi we are given is 22 over 7. The area of a circle is pi times r times r. So we need r. What we have is a diameter. 14 divided by 2. The radius is 7. Look at the 4 quarter circles in one square. One. Two. Three. Four. Four quarters of the same radius, and together they add up to one whole circle. Piece them together. It just fits the square. So write them as one circle. The area of 1 circle is 22 over 7, times 7, times 7. Cross out the 7 under the line, and the first 7. That leaves 22 times 7. That's one hundred and fifty four square centimetres. The second square has 4 more. Piece these together too. Now we have 2 circles. The area of 2 circles is 2 times one hundred and fifty four. That's three hundred and eight square centimetres. The circles are done. Now the rectangle. The area of a rectangle is length times breadth. So that's 40 times 14. That's five hundred and sixty square centimetres. Now the shaded part. It is the rectangle with the 2 circles taken out. So take one from the other. The area of the shaded part is five hundred and sixty minus three hundred and eight. Two hundred and fifty two square centimetres.

One more, before your turn. Four identical semicircles in a row. They stand on line PQ. Each semicircle sits on its own diameter, and that diameter, d, is 28 centimetres. Here it is, marked on the line. Each one overlaps the next, and the 3 overlaps are blue. The unshaded part is six hundred and twenty square centimetres more than the shaded part. Take pi as 22 over 7, and find the area of the shaded part. Look at the line, PQ. It says where they sit. It says nothing about their area. Leave it alone. Now add all 4 semicircles up. The white parts, the plain paper inside the semicircles, are covered once. But the blue parts, A, B and C, are covered twice, because 2 semicircles lie on top of each other there. So adding all 4 counts the white part once, and counts the blue part twice. So work it out. Area is pi times r times r. The radius is 28 divided by 2, so 14. The area of 1 semicircle is half, times 22 over 7, times 14, times 14. Cross out the 2 and the 7 under the lines, and the first 14 becomes 1. That leaves 22 times 14. Three hundred and eight square centimetres. The area of 4 semicircles is 4 times three hundred and eight. One thousand, two hundred and thirty two square centimetres. Look at those 4 areas on their own. Here they are, pulled apart. Each one is three hundred and eight. Now slide them back onto the line. Where 2 of them meet, a second layer goes over the blue. A, B and C each end up with 2 layers on them. So adding the 4 areas gives the whole figure once, and the blue part a second time. The question calls the white part unshaded and the blue part shaded. So the 4 semicircles are the unshaded part, plus the shaded part, plus the shaded part again. Now give the shaded part a name. A, B and C together are the shaded part, and we call that 1 unit. Draw it as bars. Here is the shaded part. 1 bar, and that bar is 1 unit. The question says the unshaded part is six hundred and twenty more than the shaded part. So the unshaded bar is the same 1 unit, and six hundred and twenty more. So to get the total area of the 4 semicircles, which is one thousand, two hundred and thirty two, remember we count the shaded part twice. So we add 1 more unit of shaded. All 3 bars together are the 4 semicircles. One thousand, two hundred and thirty two. Read it off the bars. 3 units plus six hundred and twenty is one thousand, two hundred and thirty two. Take the six hundred and twenty off the end, and 3 units are left. That is one thousand, two hundred and thirty two minus six hundred and twenty. Six hundred and twelve. Share that between the 3 units. 1 unit is six hundred and twelve divided by 3. Two hundred and four. And 1 unit is the shaded part. So the area of the shaded part is two hundred and four square centimetres.

Your turn. A rectangle is 45 centimetres long and 12 centimetres wide. Three squares in a row inside it. Each square has a side as long as the width of the rectangle, so each one reaches from the top edge to the bottom edge. Here is the first. Here is the second. And here is the third. Now look at the first square on its own. One semicircle stands on the top side and bulges down into the square. And one stands on the bottom side and bulges up. The 2 of them meet at the middle of the square. The other 2 squares are drawn the same way. All 6 semicircles are shaded, so all 6 are blue. Find the area that is NOT shaded. Everything the blue leaves. Take pi as three point one four. Try it first. Pause if you like. The rectangle first. The area of the rectangle is 45 times 12. Five hundred and forty square centimetres. Now walk the length down. Look at all 4 sides of 1 square. The length of the square is the width of the rectangle, 12. Each semicircle stands on a side of that square, so 12 is its diameter. Halve it. The radius is 6, and there it is on the diagram. The 2 semicircles in one square meet at the middle and nowhere else. Swing them together. One circle. Same rule first. Area is pi times r times r. So the area of 1 circle is three point one four, times 6, times 6. One hundred and thirteen point zero four square centimetres. 3 squares, so we have 3 circles. The area of 3 circles is 3 times one hundred and thirteen point zero four gives us three hundred and thirty nine point one two square centimetres. Read the question again. It wants the part that is not shaded. So take the blue off the whole rectangle. The area of the unshaded part is five hundred and forty minus three hundred and thirty nine point one two equals two hundred point eight eight square centimetres.