Grade 7 · Grade 7: Scale, Circles and Angles · Scale, Circles and Angles
Circles, Angles and Composite Measurement
Derive circumference and area of circles, and use angle relationships to find unknown measures.
Learning objectives
- Explain why C = πd and use A = πr².
- Use supplementary, complementary, vertical and adjacent angles to write equations.
- Solve area, surface area and volume problems for composite figures.
AERO Mathematics alignment
AERO.M7.GM.4
Know and use the formulas for the area and circumference of a circle and explain the relationship between them.
AERO.M7.GM.5
Use facts about supplementary, complementary, vertical and adjacent angles in multi-step problems to write and solve simple equations for an unknown angle.
AERO.M7.GM.6
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms.
MYP criteria
- Criterion A — Knowing and understanding
- Criterion B — Investigating patterns
- Criterion C — Communicating
- Criterion D — Applying mathematics in real-life contexts
ENGAGE
Start here
Wrap a string around a circular plate and measure both the distance around it and the distance across it.
Think about it
Divide the distance around by the distance across. Why does every circle give roughly the same answer?
Hint: That constant ratio has a name.
EXPLAIN
Circles and angle relationships
For every circle, circumference divided by diameter equals π ≈ 3.14159. That gives C = πd = 2πr. The area formula A = πr² comes from cutting the circle into thin sectors and rearranging them into an approximate rectangle of height r and width half the circumference.
| Angle pair | Relationship |
|---|---|
| Complementary | Add to 90° |
| Supplementary | Add to 180° |
| Vertical (opposite) | Equal |
| Angles on a straight line | Add to 180° |
| Angles around a point | Add to 360° |
Writing equations for unknown angles
If two supplementary angles are (2x + 10)° and (3x)°, then 2x + 10 + 3x = 180, so 5x = 170 and x = 34.
Composite figures
Break a composite shape into circles, semicircles, rectangles and triangles, calculate each area, then add or subtract. The same idea works for surface area and volume of composite solids.
Common misconception
Doubling the radius does not double the area. Because area depends on r², doubling the radius multiplies the area by four.
INVESTIGATION
Try it yourself
Measure the circumference and diameter of five circular objects. Record the ratio each time in a table, average your results and compare with π. Explain any measurement error using percent error.
Watch
Circles, Circumference and Area
Math Antics · 10:00
Explains where π comes from and how C and A are related.
Before you watch: How many diameters fit around a circle?
- Find the circumference of a circle with radius 7 cm.
- Two angles on a straight line: one is 118°. Find the other.
Interactive simulation · PhET
Function Builder
Build a machine that turns a radius into a circumference, then into an area.
While you explore
- Which operation turns r into C? Which turns r into A?
- Why does doubling the radius multiply the area by four?
Key vocabulary
- Vertical angles
- Opposite angles formed when two lines cross.
- Pi
- The number you get when you divide any circumference by its diameter.
Practice questions
0/1 correct
Level 1 · Criterion A
A circle has radius 5 cm. What is its area, to one decimal place?
MYP criterion tasks
Level 2 · Criterion C
Two supplementary angles are (2x + 20)° and (3x)°. Find x and both angles.
Reflect & track
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