Praxis Core Geometry Summary

1. Add the given angles: 47° + 68° = 115°.
2. Third angle: 180° − 115° = 65°.

Check: 47 + 68 + 65 = 180.

Wrong choices

• "115°": 115° is the sum of the two given angles, not the third angle.
• "90°": Nothing says this is a right triangle.
• "245°": This subtracts from 360°. The angles of a triangle add up to 180°, not 360°.
• "21°": This is the difference between the two given angles.

1. Area = 9 × 4.
2. Area = 36 square feet.

Check: 9 rows of 4 one-foot squares make 36 squares.

Wrong choices

• "26 square feet": 26 is the perimeter (9 + 4 + 9 + 4), which is measured in feet, not square feet.
• "13 square feet": This adds length and width instead of multiplying.
• "72 square feet": This doubles the area. The area of a rectangle is just length × width.
• "81 square feet": This squares the length (9 × 9) and ignores the width.

1. 9^2 + 12^2 = 81 + 144 = 225.
2. c = square root of 225 = 15 inches.

Check: 9, 12, 15 is 3 times the familiar 3-4-5 right triangle.

Wrong choices

• "21 inches": This adds the two legs. The hypotenuse is shorter than the sum of the legs.
• "about 7.9 inches": This subtracts the squares (144 − 81). The hypotenuse is the LONGEST side, so the squares are added.
• "225 inches": 225 is the square of the hypotenuse. Take the square root to get 15.
• "10.5 inches": This averages the legs; the Pythagorean theorem does not work that way.

1. Angles on a straight line add up to 180°.
2. 180° − 132° = 48°.

Check: 132 + 48 = 180.

Wrong choices

• "42°": This subtracts 90° from 132°, as if the angles formed a right angle.
• "228°": This subtracts from 360°. Angles on a straight line add up to 180°.
• "132°": Equal angles come from vertical angles, not from angles that form a straight line.
• "90°": Only a straight line split by a perpendicular line gives two 90° angles.

1. Diameter = 2 × 5 = 10 cm.
2. Circumference = 3.14 × 10 = 31.4 cm.

Check: a circle's circumference is a little more than 3 times its diameter; 3 × 10 = 30.

Wrong choices

• "78.5 cm": 78.5 is the AREA (π × 5 × 5), and area is in square centimeters.
• "15.7 cm": This uses π × radius. Circumference is π × diameter, or 2 × π × radius.
• "62.8 cm": This uses 2 × π × diameter, which doubles the answer.
• "157 cm": This multiplies 3.14 by 50 (a decimal-place slip).

1. Volume = 8 × 5 × 3.
2. 8 × 5 = 40, and 40 × 3 = 120 cubic inches.

Check: the base holds 40 one-inch cubes, and 3 layers make 120.

Wrong choices

• "158 cubic inches": 158 is the surface area (the area of all six faces), measured in square inches.
• "40 cubic inches": This multiplies only two of the three measurements.
• "16 cubic inches": This adds the measurements instead of multiplying.
• "240 cubic inches": This doubles the volume, as if counting two layers.

1. The person's height is 6 ÷ 4 = 1.5 times the shadow.
2. Tree height = 1.5 × 22 = 33 feet.

Check: 6/4 = 33/22, since both equal 1.5.

Wrong choices

• "about 14.7 feet": This sets the ratio upside down (22 × 4 ÷ 6). The tree's height must be greater than its shadow, just as the person's is.
• "24 feet": This adds the 2-foot difference between the person and the shadow. Similar triangles use multiplication, not adding.
• "26 feet": This adds 4 to the tree's shadow, which has no meaning here.
• "132 feet": This multiplies 22 by 6 but forgets to divide by 4.

1. Rectangle: 10 × 6 = 60 square feet.
2. Triangle: 1/2 × 10 × 4 = 20 square feet.
3. Total: 60 + 20 = 80 square feet.

Check: the triangle is half of a 10 × 4 rectangle (40), so 20 is right.

Wrong choices

• "100 square feet": This forgets that a triangle's area is HALF of base × height.
• "84 square feet": This uses 6 feet as the triangle's base. The triangle sits on the 10-foot edge.
• "64 square feet": This adds the triangle's height (4) instead of its area.
• "240 square feet": This multiplies all three numbers, which would be a volume, not an area.

1. Area = π × r^2.
2. New area = π × (3r)^2 = π × 9r^2 = 9 × (π × r^2).

Check: r = 1 gives area π; r = 3 gives area 9π, which is 9 times as large.

Wrong choices

• "3": The radius triples, but area depends on the radius SQUARED.
• "6": This doubles the factor 3 instead of squaring it.
• "27": 27 is 3 cubed. That would apply to the volume of a sphere, not the area of a circle.
• "3π": π appears in the area formula for both circles, so it cancels when comparing them.

1. Radius squared: 3 × 3 = 9.
2. Base area: 3.14 × 9 = 28.26 square inches.
3. Volume: 28.26 × 10 = 282.6 cubic inches.

Check: 9 × 10 = 90, and 90 × 3.14 = 282.6.

Wrong choices

• "188.4 cubic inches": This uses 2 × radius (6) instead of radius squared (9).
• "94.2 cubic inches": This uses the radius (3) instead of the radius squared (9).
• "1,130.4 cubic inches": This squares the diameter (6 × 6 = 36) instead of the radius.
• "90 cubic inches": This leaves out π.