Grade 7 Geometry: Angles, Area, Volume & Shapes Explained (US Common Core)

GRADE 7 MATHEMATICS • LESSON 8

Geometry

الهندسة

Name and measure angles, use the key angle facts, and find the perimeter and area of rectangles, triangles, parallelograms and circles — then take 15 exams with full step-by-step solutions.

Angles and their types

An angle measures the amount of turn between two rays, in degrees (°). The two sizes to remember are 90° (a right angle, a square corner) and 180° (a straight line).

Drag the slider to turn the angle and see what it is called, along with its complement and supplement.

📐 Angle Explorer
Angle
Type
Complement (to 90°)
Supplement (to 180°)

Words you must know

The exact names and facts you will be tested on.

Acute angle
An angle smaller than 90°.
Right angle
Exactly 90° — a square corner.
Obtuse angle
Bigger than 90° but smaller than 180°.
Straight angle
Exactly 180° — a straight line.
Reflex angle
Bigger than 180° but smaller than 360°.
ab
Complementary
Two angles that add up to 90°.
ab
Supplementary
Two angles that add up to 180°.
Perimeter
The total distance around the edge of a shape.
Area
The amount of flat space a shape covers, in square units.

Working with geometry

Your main study reference. Each fact and formula is broken into numbered steps, each one explained, with a worked example and a teacher hack.

1) The angle facts you must know

aba + b = 180°
Straight line = 180°
sum = 360°
Around a point = 360°
sum = 180°
Triangle = 180°
opposite = equal
Vertically opposite
  1. Angles on a straight line add up to 180°.
  2. Angles around a point add up to 360°.
  3. Angles in a triangle add up to 180°.
  4. Vertically opposite angles (where two lines cross) are equal.
Example: a triangle has angles 50° and 60°. Find the third.
180 − 50 − 60 = 70°
Hack: to find a missing angle, decide which fact applies (line = 180, point = 360, triangle = 180), then subtract what you know.

2) Complement and supplement

a + b = 90°
Complement (90°)
a + b = 180°
Supplement (180°)
  1. Complement: what is left to reach 90°. Subtract the angle from 90.
  2. Supplement: what is left to reach 180°. Subtract the angle from 180.
Example: the complement and supplement of 30°
Complement: 90 − 30 = 60°
Supplement: 180 − 30 = 150°
Hack: Complement goes with the smaller number (90); Supplement with the bigger (180). C comes before S, 90 before 180.

3) Perimeter and area of a rectangle

lengthwidth
Length × Width
  1. Perimeter is the distance around: add all four sides, or use 2 × (length + width).
  2. Area is the space inside: length × width.
Example: a rectangle 8 by 5
Perimeter: 2 × (8 + 5) = 26 units
Area: 8 × 5 = 40 square units
Hack: perimeter is a length (units); area is a covering (square units). Keep the units straight.

4) Area of a triangle and parallelogram

baseheight
Triangle: (b×h)÷2
baseheight
Parallelogram: b×h
  1. Triangle: area = (base × height) ÷ 2.
  2. Parallelogram: area = base × height (no dividing by 2).
  3. Height is always the straight up-and-down distance, not a slanted side.
Example: a triangle with base 10 and height 6
(10 × 6) ÷ 2 = 60 ÷ 2 = 30 square units
Hack: a triangle is exactly half of a rectangle around it — that is why you divide by 2.

5) Circumference and area of a circle

rd
radius r, diameter d
  1. Circumference (distance around) = 2 × π × radius.
  2. Area = π × radius × radius.
  3. Use π ≈ 3.14 unless told otherwise.
Example: a circle with radius 5
Circumference: 2 × 3.14 × 5 = 31.4 units
Area: 3.14 × 5 × 5 = 78.5 square units
Hack: the radius is half the diameter. If you are given the diameter, halve it first.
🧮 Shape Calculator

Test Yourself — 15 Exams

Each exam has 10 questions, and every answer comes with a step-by-step solution — even when you get it right.

Key takeaways

1. Angles: acute (<90), right (90), obtuse (90–180), straight (180), reflex (180–360).

2. Straight line = 180°, around a point = 360°, triangle = 180°.

3. Complement adds to 90°; supplement adds to 180°.

4. Rectangle: area = l × w, perimeter = 2(l + w). Triangle area = (base × height) ÷ 2.

5. Circle: circumference = 2πr, area = πr², with π ≈ 3.14.