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Praxis Core · Mathematics (5733)

Praxis Core: Geometry Concepts

Importance ★★★★☆ · 14% (estimate)

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Written by AI. Only units that passed a separate AI check (re-reading facts and numbers) are published. Tell us if you find a mistake.

Overview

This chapter covers shapes and space. You will use angle facts, congruence and similarity, the Pythagorean theorem, and formulas for perimeter, area, and volume.

The angles of a triangle add up to 180 degrees, and angles along a straight line also add up to 180 degrees. Similar figures have the same shape, so their angles are equal and their sides are proportional. Congruent figures have the same shape and the same size.

The Pythagorean theorem finds a missing side of a right triangle. Perimeter measures the distance around a shape, area measures the surface inside it in square units, and volume measures the space inside a solid in cubic units.

Key concepts

  • Angle facts

    The angles of a triangle add up to 180°. Angles on a straight line add up to 180°. Angles all the way around a point add up to 360°. A right angle is 90°.

    Memory tipTriangle and straight line: 180. Full turn: 360.

  • Pythagorean theorem

    In a right triangle, a² + b² = c², where c is the hypotenuse, the longest side across from the right angle. A common example is 3, 4, 5.

    Memory tipOnly for right triangles; c is always the longest side.

  • Similar vs. congruent

    Similar figures have equal angles and proportional sides (same shape, any size). Congruent figures have equal angles and equal sides (same shape, same size).

    Memory tipSimilar = scaled copy. Congruent = exact copy.

  • Area formulas

    Rectangle: length × width. Triangle: ½ × base × height. Circle: π × r². Area is measured in square units.

    Memory tipTriangle is half a rectangle.

  • Perimeter and circumference

    Perimeter is the sum of all side lengths. The distance around a circle is its circumference: 2 × π × r, or π × diameter.

    Memory tipPerimeter = walk around the edge.

  • Volume formulas

    Rectangular box: length × width × height. Cylinder: π × r² × height. Volume is measured in cubic units.

    Memory tipVolume = area of the base × height.

Often tested

  • The angles of a triangle add up to 180 degrees.
  • Similar figures have equal angles and proportional sides.
  • Area uses square units; volume uses cubic units.
  • In a² + b² = c², c is the hypotenuse across from the right angle.
  • Radius is half the diameter; check which one the problem gives.

Easy to confuse: Perimeter vs. area vs. volume

MeasureWhat it measuresUnits
PerimeterDistance around a flat shapeUnits (ft, cm)
AreaSurface inside a flat shapeSquare units (ft², cm²)
VolumeSpace inside a solidCubic units (ft³, cm³)

3 sample questions

Verified practice questions from this unit.

Practice questionReviewedMathematics (5733) › Geometry★★★★☆

Question 1. Two angles of a triangle measure 47° and 68°. What is the measure of the third angle?

  1. ① 245°
  2. ② 21°
  3. ③ 65°
  4. ④ 115°
  5. ⑤ 90°
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Answer: ③ 65°

Key point: The three angles of any triangle add up to 180°.

  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.
Practice questionReviewedMathematics (5733) › Geometry★★★★☆

Question 2. A rectangular rug is 9 feet long and 4 feet wide. What is its area?

  1. ① 81 square feet
  2. ② 26 square feet
  3. ③ 13 square feet
  4. ④ 72 square feet
  5. ⑤ 36 square feet
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Answer: ⑤ 36 square feet

Key point: Area of a rectangle = length × width (in square units).

  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.
Practice questionReviewedMathematics (5733) › Geometry★★★★☆

Question 3. A right triangle has legs of 9 inches and 12 inches. How long is the hypotenuse?

  1. ① 225 inches
  2. ② 10.5 inches
  3. ③ 21 inches
  4. ④ 15 inches
  5. ⑤ about 7.9 inches
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Answer: ④ 15 inches

Key point: Pythagorean theorem: a^2 + b^2 = c^2, where c is the hypotenuse.

  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.
Practice 5 questions from this unit →