Circles, Area, Volume and Scale Drawings: Grade 7 Study Guide
Geometry in Grade 7 connects formulas to real objects, from pizzas to swimming pools.
The circumference of a circle is the distance around it: C = 2πr = πd. The area is A = πr². Pi (π) is the ratio of any circle's circumference to its diameter, approximately 3.14 or 22/7. If a circle has a radius of 5 cm, then C ≈ 2 × 3.14 × 5 = 31.4 cm and A ≈ 3.14 × 25 = 78.5 cm².
The volume of a prism is the area of its base times its height: V = Bh. Surface area is the total area of all faces. Scale drawings, such as maps and blueprints, use a ratio to represent real lengths.
Key formulas
| Shape | Formula | Example |
|---|---|---|
| Circle circumference | C = 2πr | r = 3 → ≈ 18.84 |
| Circle area | A = πr² | r = 3 → ≈ 28.26 |
| Triangle area | ½ × b × h | b = 10, h = 4 → 20 |
| Prism volume | V = Bh | B = 12, h = 5 → 60 |
| Cube surface area | 6s² | s = 3 → 54 |
Angles
Angles on a straight line add to 180° (supplementary); angles that add to 90° are complementary; vertical angles (opposite each other where two lines cross) are equal. The angles in a triangle add up to 180°. Use these facts to write and solve equations for unknown angles: x + 2x + 60 = 180 → 3x = 120 → x = 40.
On a sketch.
Straight line, vertical, triangle.
Sum equals 180° or 90°.
Do the angles add up?
Scale drawings and cross sections
On a floor plan with a scale of 1 cm = 2 m, a room 4.5 cm wide is 9 m wide. The area scale is the square of the length scale: doubling every side makes the area four times as big. Slicing a prism parallel to its base gives a cross section the same shape as the base; slicing across gives a rectangle.
Worked examples and common mistakes
Example: A circular pool has a diameter of 8 m. Radius 4 m, area = 3.14 × 16 = 50.24 m². Common mistakes: using the diameter in the area formula, mixing up circumference and area, and forgetting that area is in square units and volume in cubic units. Write the formula before you substitute.
Check your understanding: model answers
1. 314 cm². 2. 90 cubic units. 3. 62°.
✨ Key Takeaways
- C = 2πr and A = πr².
- Volume of a prism = base area × height.
- Straight-line angles add to 180°.
- Area scales by the square of the length scale.
💬 Questions for Discussion & Reflection
- Find the area of a circle with radius 10 cm (use 3.14).
- Find the volume of a prism with base area 15 and height 6.
- Two angles on a straight line: one is 118°. Find the other.