GCSE Maths: Exact answers with π (Foundation)

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Put the numbers into the formula but leave π as the symbol π. Then the answer is exact.

1. Write the formula for the area of a circle. The area of a circle is π times the radius squared.
   A = πr²
2. Substitute the radius. Replace r with the radius. Leave π as the symbol.
   A = π × 3²
3. Work out the square first. In BIDMAS, powers come before multiplication. Square the radius, not the π.
   3² = 3 × 3 = 9
4. Write the number in front of π. Write the exact answer with the number first and then π, and add the units.
   9π cm²

Check. Use the π button to get a decimal and compare: 9π = 28.274 to 3 decimal places, so about 28.3 cm² to 1 d.p. This is similar in size to 28.26 found using 3.14. ✓

Watch out. Do not replace π by 3.14 or press the π key if the question says "in terms of π" or "give your answer exactly". Also square only the radius.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Put the numbers into the formula but leave π as the symbol π. Then the answer is exact.

1. Write the formula for the area of a circle. The area of a circle is π times the radius squared.
   A = πr²
2. Substitute the radius. Replace r with the radius. Leave π as the symbol.
   A = π × 5²
3. Work out the square first. In BIDMAS, powers come before multiplication. Square the radius, not the π.
   5² = 5 × 5 = 25
4. Write the number in front of π. Write the exact answer with the number first and then π, and add the units.
   25π cm²

Check. Use the π button to get a decimal and compare: 25π = 78.54 to 3 decimal places, so about 78.5 cm² to 1 d.p. This is similar in size to 78.5 found using 3.14. ✓

Watch out. Do not replace π by 3.14 or press the π key if the question says "in terms of π" or "give your answer exactly". Also square only the radius.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Put the numbers into the formula but leave π as the symbol π. Then the answer is exact.

1. Write the formula for the area of a circle. The area of a circle is π times the radius squared.
   A = πr²
2. Substitute the radius. Replace r with the radius. Leave π as the symbol.
   A = π × 6²
3. Work out the square first. In BIDMAS, powers come before multiplication. Square the radius, not the π.
   6² = 6 × 6 = 36
4. Write the number in front of π. Write the exact answer with the number first and then π, and add the units.
   36π cm²

Check. Use the π button to get a decimal and compare: 36π = 113.097 to 3 decimal places, so about 113.1 cm² to 1 d.p. This is similar in size to 113.04 found using 3.14. ✓

Watch out. Do not replace π by 3.14 or press the π key if the question says "in terms of π" or "give your answer exactly". Also square only the radius.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Put the numbers into the formula but leave π as the symbol π. Then the answer is exact.

1. Write the formula for the area of a circle. The area of a circle is π times the radius squared.
   A = πr²
2. Substitute the radius. Replace r with the radius. Leave π as the symbol.
   A = π × 10²
3. Work out the square first. In BIDMAS, powers come before multiplication. Square the radius, not the π.
   10² = 10 × 10 = 100
4. Write the number in front of π. Write the exact answer with the number first and then π, and add the units.
   100π cm²

Check. Use the π button to get a decimal and compare: 100π = 314.159 to 3 decimal places, so about 314.2 cm² to 1 d.p. This is similar in size to 314 found using 3.14. ✓

Watch out. Do not replace π by 3.14 or press the π key if the question says "in terms of π" or "give your answer exactly". Also square only the radius.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Put the numbers into the formula but leave π as the symbol π. Then the answer is exact.

1. Find the radius. The radius is half of the diameter.
   radius = 8 ÷ 2 = 4 cm
2. Write the formula for the circumference. The circumference is the distance around the circle: π times the diameter, which is also 2 × π × radius.
   C = πd   or   C = 2πr
3. Substitute. Use the diameter, which is double the radius. Leave π as the symbol.
   d = 2 × 4 = 8 cm
   C = π × 8
4. Write the number in front of π. Write the exact answer with the number first and then π, and add the units.
   8π cm

Check. Use the π button to get a decimal and compare: 8π = 25.133 to 3 decimal places, so about 25.1 cm to 1 d.p. This is similar in size to 25.12 found using 3.14. ✓

Watch out. Do not replace π by 3.14 or press the π key if the question says "in terms of π" or "give your answer exactly". Also square only the radius.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Put the numbers into the formula but leave π as the symbol π. Then the answer is exact.

1. Find the radius. The radius is half of the diameter.
   radius = 10 ÷ 2 = 5 cm
2. Write the formula for the circumference. The circumference is the distance around the circle: π times the diameter, which is also 2 × π × radius.
   C = πd   or   C = 2πr
3. Substitute. Use the diameter, which is double the radius. Leave π as the symbol.
   d = 2 × 5 = 10 cm
   C = π × 10
4. Write the number in front of π. Write the exact answer with the number first and then π, and add the units.
   10π cm

Check. Use the π button to get a decimal and compare: 10π = 31.416 to 3 decimal places, so about 31.4 cm to 1 d.p. This is similar in size to 31.4 found using 3.14. ✓

Watch out. Do not replace π by 3.14 or press the π key if the question says "in terms of π" or "give your answer exactly". Also square only the radius.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Put the numbers into the formula but leave π as the symbol π. Then the answer is exact.

1. Find the radius. The radius is half of the diameter.
   radius = 14 ÷ 2 = 7 cm
2. Write the formula for the circumference. The circumference is the distance around the circle: π times the diameter, which is also 2 × π × radius.
   C = πd   or   C = 2πr
3. Substitute. Use the diameter, which is double the radius. Leave π as the symbol.
   d = 2 × 7 = 14 cm
   C = π × 14
4. Write the number in front of π. Write the exact answer with the number first and then π, and add the units.
   14π cm

Check. Use the π button to get a decimal and compare: 14π = 43.982 to 3 decimal places, so about 44 cm to 1 d.p. This is similar in size to 43.96 found using 3.14. ✓

Watch out. Do not replace π by 3.14 or press the π key if the question says "in terms of π" or "give your answer exactly". Also square only the radius.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Put the numbers into the formula but leave π as the symbol π. Then the answer is exact.

1. Find the radius. The radius is half of the diameter.
   radius = 20 ÷ 2 = 10 cm
2. Write the formula for the circumference. The circumference is the distance around the circle: π times the diameter, which is also 2 × π × radius.
   C = πd   or   C = 2πr
3. Substitute. Use the diameter, which is double the radius. Leave π as the symbol.
   d = 2 × 10 = 20 cm
   C = π × 20
4. Write the number in front of π. Write the exact answer with the number first and then π, and add the units.
   20π cm

Check. Use the π button to get a decimal and compare: 20π = 62.832 to 3 decimal places, so about 62.8 cm to 1 d.p. This is similar in size to 62.8 found using 3.14. ✓

Watch out. Do not replace π by 3.14 or press the π key if the question says "in terms of π" or "give your answer exactly". Also square only the radius.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Put the numbers into the formula but leave π as the symbol π. Then the answer is exact.

1. Write the formula for the circumference. The circumference is the distance around the circle: π times the diameter, which is also 2 × π × radius.
   C = πd   or   C = 2πr
2. Substitute. Use the diameter, which is double the radius. Leave π as the symbol.
   d = 2 × 4 = 8 cm
   C = π × 8
3. Write the number in front of π. Write the exact answer with the number first and then π, and add the units.
   8π cm

Check. Use the π button to get a decimal and compare: 8π = 25.133 to 3 decimal places, so about 25.1 cm to 1 d.p. This is similar in size to 25.12 found using 3.14. ✓

Watch out. Do not replace π by 3.14 or press the π key if the question says "in terms of π" or "give your answer exactly". Also square only the radius.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Put the numbers into the formula but leave π as the symbol π. Then the answer is exact.

1. Write the formula for the circumference. The circumference is the distance around the circle: π times the diameter, which is also 2 × π × radius.
   C = πd   or   C = 2πr
2. Substitute. Use the diameter, which is double the radius. Leave π as the symbol.
   d = 2 × 7 = 14 cm
   C = π × 14
3. Write the number in front of π. Write the exact answer with the number first and then π, and add the units.
   14π cm

Check. Use the π button to get a decimal and compare: 14π = 43.982 to 3 decimal places, so about 44 cm to 1 d.p. This is similar in size to 43.96 found using 3.14. ✓

Watch out. Do not replace π by 3.14 or press the π key if the question says "in terms of π" or "give your answer exactly". Also square only the radius.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Put the numbers into the formula but leave π as the symbol π. Then the answer is exact.

1. Write the formula for the circumference. The circumference is the distance around the circle: π times the diameter, which is also 2 × π × radius.
   C = πd   or   C = 2πr
2. Substitute. Use the diameter, which is double the radius. Leave π as the symbol.
   d = 2 × 9 = 18 cm
   C = π × 18
3. Write the number in front of π. Write the exact answer with the number first and then π, and add the units.
   18π cm

Check. Use the π button to get a decimal and compare: 18π = 56.549 to 3 decimal places, so about 56.5 cm to 1 d.p. This is similar in size to 56.52 found using 3.14. ✓

Watch out. Do not replace π by 3.14 or press the π key if the question says "in terms of π" or "give your answer exactly". Also square only the radius.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Treat π as if it were a letter. Work with the numbers in front of it and keep π in the answer.

1. Treat π like a letter in algebra. It is a number, but we do not write it as a decimal. So 3π + 5π works just like 3x + 5x = 8x: 3 lots of π and 5 lots of π make 8 lots of π.
   3π + 5π
2. Add the numbers in front of π. The number in front tells us how many lots of π there are.
   3 + 5 = 8
3. Write the answer with π.
   8π

Check. Use 3.14 for π as a rough check: 3 lots of 3.14 and the answer 8 lots of 3.14 = 25.12, which has the right size. ✓

Watch out. π does not change when you add or subtract numbers in front of it. 3π + 5π = 8π, not 8π².

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Treat π as if it were a letter. Work with the numbers in front of it and keep π in the answer.

1. Treat π like a letter in algebra. It is a number, but we do not write it as a decimal. So 3π + 5π works just like 3x + 5x = 8x: 3 lots of π and 5 lots of π make 8 lots of π.
   9π − 4π
2. Subtract the numbers in front of π. The number in front tells us how many lots of π there are.
   9 − 4 = 5
3. Write the answer with π.
   5π

Check. Use 3.14 for π as a rough check: 9 lots of 3.14 and the answer 5 lots of 3.14 = 15.7, which has the right size. ✓

Watch out. π does not change when you add or subtract numbers in front of it. 3π + 5π = 8π, not 8π².

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Treat π as if it were a letter. Work with the numbers in front of it and keep π in the answer.

1. Treat π like a letter in algebra. It is a number, but we do not write it as a decimal. So 3π + 5π works just like 3x + 5x = 8x: 3 lots of π and 5 lots of π make 8 lots of π.
   12π − π
2. Subtract the numbers in front of π. The number in front tells us how many lots of π there are.
   12 − 1 = 11
3. Write the answer with π.
   11π

Check. Use 3.14 for π as a rough check: 12 lots of 3.14 and the answer 11 lots of 3.14 = 34.54, which has the right size. ✓

Watch out. π does not change when you add or subtract numbers in front of it. 3π + 5π = 8π, not 8π².

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Treat π as if it were a letter. Work with the numbers in front of it and keep π in the answer.

1. Treat π like a letter in algebra. It is a number, but we do not write it as a decimal. So 3π + 5π works just like 3x + 5x = 8x: 3 lots of π and 5 lots of π make 8 lots of π.
   7π + 2π
2. Add the numbers in front of π. The number in front tells us how many lots of π there are.
   7 + 2 = 9
3. Write the answer with π.
   9π

Check. Use 3.14 for π as a rough check: 7 lots of 3.14 and the answer 9 lots of 3.14 = 28.26, which has the right size. ✓

Watch out. π does not change when you add or subtract numbers in front of it. 3π + 5π = 8π, not 8π².

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Treat π as if it were a letter. Work with the numbers in front of it and keep π in the answer.

1. Treat π like a letter in algebra. 3π × 4 works like 3x × 4 = 12x. Multiply the numbers; π stays where it is.
   4π × 3
2. Multiply the numbers. Do not turn π into 3.14.
   4 × 3 = 12
3. Write the answer with π.
   12π

Check. Use 3.14 for π as a rough check: 4 lots of 3.14 and the answer 12 lots of 3.14 = 37.68, which has the right size. ✓

Watch out. π does not change when you add or subtract numbers in front of it. 3π + 5π = 8π, not 8π².

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Treat π as if it were a letter. Work with the numbers in front of it and keep π in the answer.

1. Treat π like a letter in algebra. 3π × 4 works like 3x × 4 = 12x. Multiply the numbers; π stays where it is.
   34π × 8
2. Multiply the numbers. Do not turn π into 3.14.
   34 × 8 = 6
3. Write the answer with π.
   6π

Check. Use 3.14 for π as a rough check: 34 lots of 3.14 and the answer 6 lots of 3.14 = 18.84, which has the right size. ✓

Watch out. π does not change when you add or subtract numbers in front of it. 3π + 5π = 8π, not 8π².

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Treat π as if it were a letter. Work with the numbers in front of it and keep π in the answer.

1. Treat π like a letter in algebra. 3π × 4 works like 3x × 4 = 12x. Multiply the numbers; π stays where it is.
   5π × 6
2. Multiply the numbers. Do not turn π into 3.14.
   5 × 6 = 30
3. Write the answer with π.
   30π

Check. Use 3.14 for π as a rough check: 5 lots of 3.14 and the answer 30 lots of 3.14 = 94.2, which has the right size. ✓

Watch out. π does not change when you add or subtract numbers in front of it. 3π + 5π = 8π, not 8π².

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Treat π as if it were a letter. Work with the numbers in front of it and keep π in the answer.

1. Treat π like a letter in algebra. 12π ÷ 4 works like 12x ÷ 4 = 3x. Divide the number in front; π stays where it is.
   6π ÷ 2
2. Divide the numbers.
   6 ÷ 2 = 3
3. Write the answer with π.
   3π

Check. Use 3.14 for π as a rough check: 6 lots of 3.14 and the answer 3 lots of 3.14 = 9.42, which has the right size. ✓

Watch out. π does not change when you add or subtract numbers in front of it. 3π + 5π = 8π, not 8π².

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Treat π as if it were a letter. Work with the numbers in front of it and keep π in the answer.

1. Treat π like a letter in algebra. 12π ÷ 4 works like 12x ÷ 4 = 3x. Divide the number in front; π stays where it is.
   10π ÷ 4
2. Divide the numbers.
   10 ÷ 4 = 52
3. Write the answer with π.
   52π

Check. Use 3.14 for π as a rough check: 10 lots of 3.14 and the answer 52 lots of 3.14 = 7.85, which has the right size. ✓

Watch out. π does not change when you add or subtract numbers in front of it. 3π + 5π = 8π, not 8π².

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Type the exact answer into the calculator using the π key and round as asked.

1. Use the π button on the calculator. The π button holds π to many more digits than 3.14, so the answer will be more accurate.
   36π = 113.097336… (calculator display)
2. Round to 1 decimal place. Look at the next digit after decimal place 1: 5 or more rounds up.
   113.1 cm²

Check. Rough check with 3: 36 × 3 = 108, which is a little below 113.1. ✓

Watch out. Round only at the very end.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Type the exact answer into the calculator using the π key and round as asked.

1. Use the π button on the calculator. The π button holds π to many more digits than 3.14, so the answer will be more accurate.
   10π = 31.415927… (calculator display)
2. Round to 1 decimal place. Look at the next digit after decimal place 1: 5 or more rounds up.
   31.4 cm

Check. Rough check with 3: 10 × 3 = 30, which is a little below 31.4. ✓

Watch out. Round only at the very end.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Type the exact answer into the calculator using the π key and round as asked.

1. Use the π button on the calculator. The π button holds π to many more digits than 3.14, so the answer will be more accurate.
   25π = 78.539816… (calculator display)
2. Round to 2 decimal places. Look at the next digit after decimal place 2: 5 or more rounds up.
   78.54 m²

Check. Rough check with 3: 25 × 3 = 75, which is a little below 78.54. ✓

Watch out. Round only at the very end.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Find the area of the whole circle, then take the fraction of it that the shape is. Leave π in the answer.

1. Write the area of the whole circle. A = πr².
   A = π × 8² = 64π cm²
2. Take the fraction of it. The shape is a quarter of the circle, so multiply by 14.
   14 × 64π = 16π
3. Write the answer with units.
   16π cm²

Check. As a decimal this is about 50.3 cm², which is less than the whole circle (201.1 cm²). ✓

Watch out. Use the radius. If you are given the diameter, halve it first.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Find the area of the whole circle, then take the fraction of it that the shape is. Leave π in the answer.

1. Write the area of the whole circle. A = πr².
   A = π × 10² = 100π cm²
2. Take the fraction of it. The shape is a quarter of the circle, so multiply by 14.
   14 × 100π = 25π
3. Write the answer with units.
   25π cm²

Check. As a decimal this is about 78.5 cm², which is less than the whole circle (314.2 cm²). ✓

Watch out. Use the radius. If you are given the diameter, halve it first.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Find the area of the whole circle, then take the fraction of it that the shape is. Leave π in the answer.

1. Write the area of the whole circle. A = πr².
   A = π × 6² = 36π cm²
2. Take the fraction of it. The shape is a half of the circle, so multiply by 12.
   12 × 36π = 18π
3. Write the answer with units.
   18π cm²

Check. As a decimal this is about 56.5 cm², which is less than the whole circle (113.1 cm²). ✓

Watch out. Use the radius. If you are given the diameter, halve it first.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Find the area of the whole circle, then take the fraction of it that the shape is. Leave π in the answer.

1. Write the area of the whole circle. A = πr².
   A = π × 4² = 16π cm²
2. Take the fraction of it. The shape is a half of the circle, so multiply by 12.
   12 × 16π = 8π
3. Write the answer with units.
   8π cm²

Check. As a decimal this is about 25.1 cm², which is less than the whole circle (50.3 cm²). ✓

Watch out. Use the radius. If you are given the diameter, halve it first.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Add the curved length (a fraction of the circumference) to the straight edges. Leave π in the answer.

1. Split the perimeter into the curved part and the straight part(s). The perimeter is the distance all the way round the shape. A semicircle has one curved edge and straight edges.
   perimeter = curved edge + straight edge(s)
2. Find the length of the curved edge. The curved edge is half of the circumference of the full circle.
   half of 2πr = πr = π × 5 = 5π cm
3. Find the length of the straight edge(s).
   the diameter = 2 × 5 = 10 cm
4. Add them. Leave π in the answer. A multiple of π and a whole number cannot be combined into one term, so the answer has two parts.
   (5π + 10) cm

Check. As a decimal: 15.71 + 10 = 25.71 cm (about). This is a sensible length for a shape with radius 5 cm. ✓

Watch out. Do not forget the straight edge(s). The perimeter of a semicircle is more than half the circumference of the circle.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Add the curved length (a fraction of the circumference) to the straight edges. Leave π in the answer.

1. Split the perimeter into the curved part and the straight part(s). The perimeter is the distance all the way round the shape. A semicircle has one curved edge and straight edges.
   perimeter = curved edge + straight edge(s)
2. Find the length of the curved edge. The curved edge is half of the circumference of the full circle.
   half of 2πr = πr = π × 6 = 6π cm
3. Find the length of the straight edge(s).
   the diameter = 2 × 6 = 12 cm
4. Add them. Leave π in the answer. A multiple of π and a whole number cannot be combined into one term, so the answer has two parts.
   (6π + 12) cm

Check. As a decimal: 18.85 + 12 = 30.85 cm (about). This is a sensible length for a shape with radius 6 cm. ✓

Watch out. Do not forget the straight edge(s). The perimeter of a semicircle is more than half the circumference of the circle.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Add the curved length (a fraction of the circumference) to the straight edges. Leave π in the answer.

1. Split the perimeter into the curved part and the straight part(s). The perimeter is the distance all the way round the shape. A quarter circle has one curved edge and straight edges.
   perimeter = curved edge + straight edge(s)
2. Find the length of the curved edge. The curved edge is a quarter of the circumference of the full circle.
   a quarter of 2πr = πr ÷ 2 = π × 8 ÷ 2 = 4π cm
3. Find the length of the straight edge(s).
   two radii = 2 × 8 = 16 cm
4. Add them. Leave π in the answer. A multiple of π and a whole number cannot be combined into one term, so the answer has two parts.
   (4π + 16) cm

Check. As a decimal: 12.57 + 16 = 28.57 cm (about). This is a sensible length for a shape with radius 8 cm. ✓

Watch out. Do not forget the straight edge(s). The perimeter of a semicircle is more than half the circumference of the circle.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Split into a rectangle and a semicircle, find each area exactly, then add.

1. Split the shape into a rectangle and a semicircle. It is easier to find the areas of simple shapes and add them.
   area = rectangle + semicircle
2. Find the area of the rectangle. area = length × width
   8 × 5 = 40 cm²
3. Find the radius of the semicircle. The diameter of the semicircle is the width of the rectangle. The radius is half the diameter.
   radius = 8 ÷ 2 = 4 cm
4. Find the area of the semicircle. A semicircle is half a circle: half of πr².
   12 × π × 4² = 12 × π × 16 = 8π cm²
5. Add the two areas. Leave π in the answer. The whole numbers and the π part cannot be combined into one term.
   (40 + 8π) cm²

Check. As a decimal: 40 + 25.13 = 65.13 cm² (about). The rectangle alone is 40 cm², so this is a little bigger. ✓

Watch out. Use the radius (half the width) in the circle formula, not the whole width.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Split into a rectangle and a semicircle, find each area exactly, then add.

1. Split the shape into a rectangle and a semicircle. It is easier to find the areas of simple shapes and add them.
   area = rectangle + semicircle
2. Find the area of the rectangle. area = length × width
   12 × 7 = 84 cm²
3. Find the radius of the semicircle. The diameter of the semicircle is the width of the rectangle. The radius is half the diameter.
   radius = 12 ÷ 2 = 6 cm
4. Find the area of the semicircle. A semicircle is half a circle: half of πr².
   12 × π × 6² = 12 × π × 36 = 18π cm²
5. Add the two areas. Leave π in the answer. The whole numbers and the π part cannot be combined into one term.
   (84 + 18π) cm²

Check. As a decimal: 84 + 56.55 = 140.55 cm² (about). The rectangle alone is 84 cm², so this is a little bigger. ✓

Watch out. Use the radius (half the width) in the circle formula, not the whole width.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Use the area formula backwards: divide by π, then take the square root.

1. Write the formula for the area.
   A = πr²
2. Put in the area and divide both sides by π. The area is 49π cm². Dividing both sides by π leaves the number.
   49π = πr², so r² = 49
3. Take the square root. 49 = 7 × 7.
   r = √49 = 7 cm

Check. Put it back: π × 7² = 49π. ✓

Watch out. Do not forget to take the square root at the end.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Use the area formula backwards: divide by π, then take the square root.

1. Write the formula for the area.
   A = πr²
2. Put in the area and divide both sides by π. The area is 100π cm². Dividing both sides by π leaves the number.
   100π = πr², so r² = 100
3. Take the square root. 100 = 10 × 10.
   r = √100 = 10 cm

Check. Put it back: π × 10² = 100π. ✓

Watch out. Do not forget to take the square root at the end.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Use the circumference formula backwards: divide by 2π.

1. Write the formula for the circumference.
   C = 2πr
2. Put in the circumference and divide by 2π. The circumference is 14π cm. Divide both sides by 2π.
   14π = 2πr, so r = 14 ÷ 2 = 7 cm

Check. Put it back: 2 × π × 7 = 14π. ✓

Watch out. C = 2πr, so r is half of the number in front of π.

Key words. π (pi) is the number 3.14159… that you get when you divide the circumference of any circle by its diameter. It never ends and never repeats, so we cannot write it exactly as a decimal. An answer left in terms of π (such as 12π) is exact: nothing has been rounded.

Method. Use the circumference formula backwards: divide by 2π.

1. Write the formula for the circumference.
   C = 2πr
2. Put in the circumference and divide by 2π. The circumference is 30π cm. Divide both sides by 2π.
   30π = 2πr, so r = 30 ÷ 2 = 15 cm

Check. Put it back: 2 × π × 15 = 30π. ✓

Watch out. C = 2πr, so r is half of the number in front of π.