Cutting a circle into wedges and relaying them makes a rectangle of pi r by r
The space inside a circle
The area of a circle is the space inside it, counted in unit squares, as for any other shape. A circle has no straight sides to multiply together, so we cut it into pieces and move the pieces into a shape we can measure.
Everything is worked out from the radius. A circle is every point that is the same distance from its center, and that distance is the radius, r.
Every point of the circle is the distance r from the center.
Cut it into wedges
Cut the circle into equal wedges from the center to the edge, like slices of a round pizza. Each wedge is called a sector. Its two straight sides are radii, and its curved edge is a piece of the circumference.
Nothing is added and nothing is thrown away. However the wedges are moved around, together they still cover exactly the area of the circle.
The circle cut into 12 equal wedges. Each one has two radii for its straight sides.
Lay them tip up, tip down
Lay the wedges in a row, one pointing up and the next pointing down, so that they fit together. The shape is nearly a rectangle, but it has bumps along the top and the bottom where the curved edges are.
Cut the circle into more wedges, each one thinner, and the bumps get smaller. The top and bottom edges straighten out, and the shape gets closer and closer to a rectangle.
Cut into 64 sectors and unfold them fully
Drag one handle to cut the circle into more sectors, and the other to lay them out. With 64 thin sectors, the shape they make is almost exactly a rectangle.
A rectangle wide and r tall
Now measure the rectangle. Its height is the straight side of one wedge, which is the radius, r.
Its top edge is made of the curved edges of the wedges that point down, and its bottom edge is made of the curved edges of the wedges that point up. Half of the circumference is along the top and half is along the bottom. The whole circumference is , so the top is half of that: .
The area of the rectangle is its width times its height: . The wedges are the whole circle, so this is the area of the circle too: area . Read as times r times r: lots of the square built on the radius.
With 40 thin wedges the shape is almost a rectangle: wide, which is half the circumference, and r tall.
Using the formula
A circle with a radius of 7 cm has an area of cm². Using for , that is cm².
When a question gives the diameter, halve it first. A circle with a diameter of 10 cm has a radius of 5 cm, so its area is , which is about 3.14 × 25 = 78.5 cm².
Double the radius
Double the radius of a circle and every length in it doubles, so its circumference doubles too. The area goes up 4 times, because the radius is used twice in r × r. A circle with a radius of 4 has an area of , which is 4 times the of a circle with a radius of 2.
Double the radius
C = 2πr is linear in r and A = πr² is quadratic, so doubling r doubles C and quadruples A
Drag the radius from 2 to 4. The circle becomes twice as wide, so its circumference is twice as long, but it covers 4 times the area: , against .
The usual mistakes
Using the diameter in place of the radius. For a circle with a diameter of 10, is 4 times the true area, which is .
Mixing up the area and the circumference. is the length round the edge, measured in units such as cm. is the space inside, measured in square units such as cm². For a radius of 7 cm, the circumference is cm and the area is cm².
Doubling instead of squaring. means r × r, not 2 × r. For a radius of 7, is 7 × 7 = 49, not 14.
Composite areas · A Circle Drawn Inside a Square
QuestionA circle of radius 7 cm is drawn inside a square and touches all four sides of the square. Take π = 227. (a) Find the total area of the four corner parts of the square outside the circle. (b) Find the area of one corner part.
1.The circle touches the left side and the right side of the square, so the side of the square is the diameter, 2 × 7 = 14 cm. The square is 14 × 14 = 196 cm2.
The circle touches the left and right sides, so the square is 2 × 7 = 14 cm across: 196 cm². 2.The area of the circle is 227 × 7 × 7 = 154 cm2.
The circle is 227 × 7 × 7 = 154 cm². 3.(a) The part of the square outside the circle is the four corners: 196 − 154 = 42 cm2.
(a) The four corners together are 196 − 154 = 42 cm². 4.The circle sits exactly in the middle of the square, so the four corner parts are the same shape and the same size.
The circle sits exactly in the middle of the square, so the four corners are the same size. 5.(b) One corner part is 42 ÷ 4 = 10.5 cm2. Check: 4 × 10.5 = 42, and 154 + 42 = 196.
(b) One corner is 42 ÷ 4 = 10.5 cm².
Answer: (a) 42 cm2; (b) 10.5 cm2
Watch out for
- Using 7 cm as the side of the square, which gives a square of 49 cm2, smaller than the circle inside it. 7 cm is the radius, from the center to a side. The circle is 14 cm across, and that is the side of the square.
- Treating a corner part as a triangle and working out 12 × 7 × 7. One of its edges is curved, so it is not a triangle. Its area can only be found by taking the circle away from the square.
Circles · A Rope Tied at a Corner
QuestionA goat is tied by a 14 m rope to the outside corner of a large shed with straight walls, longer than 14 m in both directions. Take π = 227. Find the area the goat can graze, and the length of the curved edge of that area.
1.Full circle area = 227 × 14 × 14 = 616 m2.
Without the shed the rope would sweep a full circle: 616 m². 2.The walls block a quarter: grazing area = 34 × 616 = 462 m2.
The corner blocks one quarter: 34 × 616 = 462 m². 3.Full circumference = 2 × 227 × 14 = 88 m.
The full circumference would be 88 m. 4.Curved edge = 34 × 88 = 66 m.
Three quarters of it: 66 m.
Answer: 462 m2; 66 m
Watch out for
- Giving a semicircle, as if the goat were tied to a flat wall rather than a corner.
- Adding the two 14 m rope lines along the walls to a perimeter that asked only for the curved edge.
Your turn
1.A circle of radius 7. What is its area?