Pre-Algebra

Intro to Geometry

Lesson

Geometry is mostly about three things: how long, how big, and how rotated.

Perimeter: the distance all the way around a shape. Add up the sides.
Area: the surface inside a flat shape, in square units.
Volume: the space inside a solid, in cubic units.

Key formulas

Rectangle area: $A = \ell \cdot w$
Rectangle perimeter: $P = 2\ell + 2w$
Square area: $A = s^2$
Square perimeter: $P = 4s$
Triangle area: $A = \tfrac{1}{2} b h$
Parallelogram area: $A = b h$
Rectangular prism volume: $V = \ell \cdot w \cdot h$
Cube volume: $V = s^3$
Circle circumference: $C = 2\pi r$
Circle area: $A = \pi r^2$

Worked example 1 — triangle area

Triangle with base $10$ and height $6$ :

A = \tfrac{1}{2}(10)(6) = 30

Worked example 2 — circle area

Circle with radius $4$ , using $\pi \approx 3.14$ :

A = 3.14 \cdot 4^2 = 50.24

Angles measure rotation in degrees.

Angle facts

Angles in a triangle sum to $180^\circ$ .
Angles on a straight line sum to $180^\circ$ (supplementary).
Angles in a right angle sum to $90^\circ$ (complementary).
Vertical angles (across an intersection) are equal.

Worked example 3 — missing angle

Triangle has angles $60^\circ$ and $80^\circ$ . The third:

180 - 60 - 80 = 40^\circ

Type just the number (no units, no degree symbol).

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\text{Perimeter of a square with side 6}

Problem 2

\text{Area of a 4-by-5 rectangle}

Problem 3

\text{Area of a square with side 7}

Problem 4

\text{Sum of the angles in any triangle (degrees)}

Practice

Standard problems matching the lesson.

Problem 5

\text{Perimeter of an 8-by-3 rectangle}

Problem 6

\text{Area of a 12-by-5 rectangle}

Problem 7

\text{Area of a triangle with base 10 and height 6}

Problem 8

\text{Area of a parallelogram with base 8 and height 5}

Problem 9

\text{Volume of a cube with edge 4}

Problem 10

\text{Volume of a 2-by-3-by-5 rectangular prism}

Problem 11

\text{Complement of a } 35^\circ \text{ angle}

Problem 12

\text{Supplement of an } 80^\circ \text{ angle}

Problem 13

\text{Triangle with angles } 60^\circ \text{ and } 80^\circ. \text{ Third?}

Problem 14

\text{Circumference, radius 5 (}\pi = 3.14\text{)}

Problem 15

\text{Area of circle radius 4 (}\pi = 3.14\text{)}

Problem 16

\text{Perimeter of a square whose area is 49}

Problem 17

\text{Area of an 8-by-12 rectangle}

Problem 18

\text{Volume of a cube with edge 5}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

\text{Linear pair: one angle } 55^\circ. \text{ Other?}

Problem 20

\text{Vertical angle to } 47^\circ

Problem 21

\text{L-shape: 6-by-4 rectangle with 2-by-2 square removed. Area?}

Problem 22

\text{Volume of a 3-by-7-by-4 rectangular prism}

Problem 23

\text{Right triangle, one acute } 32^\circ. \text{ Other acute?}

Problem 24

\text{Area of circle radius 10 (}\pi = 3.14\text{)}

Problem 25

\text{Volume of a 25-by-10-by-5 rectangular prism}

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