Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Every circle, from a coin to a Ferris wheel, has the same relationship between the distance around it and the distance across it: the circumference is about 3.14 times the diameter. That constant is π, and seventh graders use it in two formulas: circumference C = πd = 2πr and area A = πr². They solve problems such as how far a wheel rolls in ten turns or how much fabric covers a round table.
The standard also asks for an informal derivation that connects the two formulas. Cut a circle into many thin sectors and arrange them point-up, point-down, and they form a shape close to a parallelogram. Its base is half the circumference, πr, and its height is the radius r, so its area is πr × r = πr². Students should be able to retell this argument, explain why π appears in both formulas, and choose the right formula for a problem by asking whether the question is about the distance around or the space inside.
A = π × d² gives four times the right area. Students should halve the diameter first and label r clearly before substituting.
Fencing a round garden needs circumference, while seeding it needs area. Asking 'around or inside?' before choosing a formula prevents this mix-up.
Some students compute (πr)². The formula squares only the radius: multiply r × r first, then multiply by π.
π is an irrational number and 3.14 is an approximation. Answers using 3.14 are estimates, which is why many problems say 'use 3.14' or 'leave your answer in terms of π'.
A circle has a radius of 5 cm. Using π ≈ 3.14, find its circumference and its area.
Answer: The circumference is about 31.4 cm and the area is about 78.5 cm².
Have students measure the circumference and diameter of several round objects with string and divide; the class results cluster around 3.1 to 3.2, which gives π a real meaning. For area, paper plates cut into 8, then 16 sectors show the parallelogram emerging more clearly as the sectors get thinner.
Problems on tests often combine a circle with other shapes, such as a semicircle on a rectangle, or ask for a radius given the area. Practicing reverse problems builds flexibility with the formulas.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 31.4 (also accepted: 31.4 in, 31.4 inches)
C = πd = 3.14 × 10 = 31.4 inches.
Answer: 28.26 (also accepted: 28.26 cm², 28.26 cm2)
A = πr² = 3.14 × 3 × 3 = 3.14 × 9 = 28.26 cm².
Answer: C) 10 m
C = 2πr, so r = C ÷ (2π) = 62.8 ÷ 6.28 = 10 m. The diameter would be 20 m.
Answer: B) Half the circumference and the radius
Half of the sectors' curved edges run along the bottom, giving a base of half the circumference, πr. The height is the radius, so the area is πr × r = πr².
Answer: A) 13 m²
A = 3.14 × 2 × 2 = 12.56 m², which rounds to 13 m².
Answer: 1570 (also accepted: 1,570, 1570 cm)
One turn covers the circumference: 3.14 × 50 = 157 cm. Ten turns cover 157 × 10 = 1,570 cm.
A full lesson with slides, activities and an exit ticket on area and circumference of a circle, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.G.B.4 with an answer key, ready in about a minute.
Make a worksheet →Turn area and circumference of a circle into a quiz students answer online that marks itself, with a class summary for you.
Build a test →They need an informal derivation, such as rearranging sectors into a parallelogram, that explains where A = πr² comes from. A formal proof is not required.
Either is a reasonable approximation of π. Many problems also accept answers in terms of π, such as 25π cm², which avoids rounding altogether.