Geometry

1 · Angles you can name

An angle is measured in degrees. A right angle is 90°, an acute angle is less than 90°, an obtuse angle is between 90° and 180°, and a straight angle is 180°.

2 · Angles on a line and around a point

Angles that sit side by side on a straight line add to 180°. If one is 65°, the other is 180 − 65 = 115°.

Angles all the way around a point add to 360°.

3 · Angles in triangles and quadrilaterals

The three angles of any triangle add to 180°. With angles of 48° and 75°, the third is 180 − 48 − 75 = 57°.

The four angles of any quadrilateral add to 360°. Mixing the two totals up is a classic trap.

4 · Kinds of triangles and quadrilaterals

Triangles: equilateral (3 equal sides, each angle 60°), isosceles (2 equal sides), scalene (no equal sides), and right (one 90° angle).

Quadrilaterals: a square is also a rectangle and a rhombus. A rectangle has 4 right angles. A rhombus has 4 equal sides. A parallelogram has 2 pairs of parallel sides. A trapezoid has exactly one pair of parallel sides.

5 · Perimeter and area of rectangles

Perimeter is the distance around: P = 2 × (length + width). Area is the space inside: A = length × width, in square units.

Going backwards: a rectangle with area 48 and length 8 has width 48 ÷ 8 = 6, so its perimeter is 2 × (8 + 6) = 28.

6 · L shapes

For area, cut the shape into two rectangles, or take the big rectangle and subtract the missing corner: 10 × 8 − 4 × 3 = 80 − 12 = 68.

For perimeter, the missing corner does not change the total: the two inside edges are just as long as the two outside pieces that were removed. Perimeter = 2 × (10 + 8) = 36.

7 · Area of a triangle

A triangle is half of a rectangle with the same base and height: A = ½ × base × height.

Base 10, height 6: ½ × 10 × 6 = 30. The height is measured straight up from the base, not along the slanted side.

8 · Volume of a box

Volume counts the cubes that fill a box: V = length × width × height. A box 5 by 4 by 3 holds 5 × 4 × 3 = 60 cubes.

Working backwards: volume 72 with a 4 by 3 base means the height is 72 ÷ 12 = 6.

9 · The coordinate grid

A point (x, y) is x steps right and y steps up from (0, 0). Order matters: (2, 5) and (5, 2) are different points.

A rectangle with corners A(1, 1), B(5, 1), and C(5, 4) has its fourth corner at D(1, 4). Its sides are 5 − 1 = 4 and 4 − 1 = 3 long, so its area is 12.

10 · Squares from the perimeter

A square with perimeter 36 has sides of 36 ÷ 4 = 9, so its area is 9 × 9 = 81.

Perimeter and area are different measures. Doubling a square's side doubles its perimeter but makes its area 4 times as big.