Geometry and measures, Year 9: circles and cylinders

388.4 m

提示Put the two curved ends together and see what familiar shape they make.

为什么The two semicircles make one full circle of diameter 60 m, with circumference 3.14 × 60 = 188.4 m. Adding the two straights gives 188.4 + 200. Splitting a shape into parts you know is the first step with any compound shape.

7.57 m²

提示Find the two parts separately, and halve the width before you use it in the curved part.

为什么The rectangle is 2 × 3 = 6 m². The semicircle has radius 1 m, so its area is 3.14 × 1² ÷ 2 = 1.57 m². The whole window is the two added together.

45.7 cm

提示Trace round the outside with a finger and note which edges you actually touch.

为什么The outline is three sides of the square, 30 cm, plus the curved edge, half of 3.14 × 10 = 15.7 cm. The fourth side of the square is where the two parts join, so it lies inside the shape.

78.5 cm²

提示Work out the full circle first, then take the fraction of it that the name tells you.

为什么A full circle of radius 10 cm has area 3.14 × 10² = 314 cm². A quarter-circle is one of four equal slices of it, so its area is 314 ÷ 4.

86 cm²

提示Find each of the two areas, then decide whether the round one is being put in or taken out.

为什么The square is 20 × 20 = 400 cm² and the circle is 3.14 × 10² = 314 cm². The circle has been removed, so the card left over is 400 − 314. Some compound shapes are built by taking a piece away.

1570 cm³

提示Find the area of the circular end first, then multiply by how far that circle is stacked.

为什么The circular end has area 3.14 × 10² = 314 cm², and the cylinder is that circle stacked 5 cm high, so the volume is 314 × 5. A cylinder is a prism: cross-section area × length.

3

提示Work out the area of the circular end, then ask how many of those layers fill the space.

为什么The end has area 3.14 × 10² = 314 cm². Volume = end area × height, so height = 942 ÷ 314. Working backwards means undoing the last multiplication.

502 cm³

提示Check which measurement across the circle you have been given before you square anything.

为什么The formula needs the radius, which is half of 8 cm. Then 3.14 × 4² × 10 = 3.14 × 160 = 502.4, which rounds to the nearest whole number.

It becomes four times as large

提示In the formula the radius is squared, so see what squaring does to a doubled number.

为什么Doubling r turns r² into (2r)² = 4r², so the circular end has four times the area. With the same height stacked on top, the volume is four times as large too.

2πrh

提示Unroll the label from a tin: a rectangle as wide as the distance round the rim.

为什么The curved surface unrolls into a rectangle. One side is the height h and the other is the circumference 2πr, so the area is 2πr × h.

The circumference of the circular end

提示It has to wrap exactly once round the rim.

为什么Peel the label off a tin and it lies flat as a rectangle. Its height is the height of the tin, and its width is exactly the distance round the tin, which is why the circumference appears in the formula.

125.6 cm²

提示Find the distance round the rim, then multiply by how tall the side is.

为什么The circumference is 2 × 3.14 × 4 = 25.12 cm, and the curved surface is a rectangle that wide and 5 cm tall: 25.12 × 5 = 125.6.

1256 cm²; 1884 cm²

提示Unroll the side into a rectangle for the first answer, then count every face of the solid for the second.

为什么The curved surface is 2 × 3.14 × 10 × 20. A closed cylinder also has a circle at the top and another at the bottom, so two lots of 314 cm² are added to it.

one

提示Picture the object and count the flat circular faces it really has.

为什么The formula 2πr² + 2πrh is for a closed cylinder. A bucket or a drinking glass has a single circle, and a tube open at both ends has none, so read what the object is before choosing which parts to add.