What the question looks like
Geometry questions on the HSPT usually come with a short description of a shape, and sometimes a small diagram. You might be given the base and height of a triangle and asked for its area, told two angles of a triangle and asked for the third, or given the three dimensions of a box and asked for its volume.
The topics include perimeter, area of rectangles, triangles, and circles, angle relationships along a straight line and inside a triangle, vertical angles, the coordinate plane, volume of a rectangular box, and units. Units matter here: an area answer will be in square units, and a volume answer in cubic units.
None of this requires a proof or the Pythagorean theorem. If a right triangle shows up, it will almost always be a 3-4-5 triangle or a multiple of one, such as 6-8-10.
What you need to know
- Perimeter is the distance around a shape. Add up every side.
- Area of a rectangle = length × width.
- Area of a triangle = one half × base × height.
- Circumference of a circle = 2 × pi × radius, or pi × diameter. Use 3.14 for pi.
- Area of a circle = pi × radius × radius.
- The diameter is twice the radius. If you are given the diameter, cut it in half before using a circle formula.
- The angles of a triangle add up to 180 degrees.
- Angles along a straight line add up to 180 degrees. Angles around a point add up to 360 degrees.
- Vertical angles, the opposite angles formed by two crossing lines, are equal.
- Volume of a rectangular box = length × width × height.
- On the coordinate plane, a point is written (x, y). The x-value tells you how far right or left, and the y-value tells you how far up or down.
- A 3-4-5 triangle is a right triangle, and so is any multiple of it, like 6-8-10 or 9-12-15.
How to solve it
- Sketch the shape if there is no diagram, and label every number you are given, including units.
- Decide what the question wants: perimeter, area, an angle, a length, or volume. Write down the matching formula.
- Substitute the values into the formula. Watch for radius versus diameter, and base versus a slanted side.
- Compute, then attach the right unit: plain units for length, square units for area, cubic units for volume.
- Check that the size makes sense. A triangle's area should be smaller than a rectangle with the same base and height.
Worked examples
Example 1
A triangle has a base of 12 cm and a height of 7 cm. What is its area?
- A.19 cm²
- B.38 cm²
- C.42 cm²
- D.84 cm²
Show answer
Answer: C: 42 cm²
Area of a triangle is one half × base × height, so one half × 12 × 7 = 6 × 7 = 42 square centimeters. Choice D, 84 cm², is base times height with the one half forgotten, which would be the area of a rectangle, not a triangle. Choice A, 19, just adds the two numbers.
Example 2
Two angles of a triangle measure 48° and 67°. What is the measure of the third angle?
- A.55°
- B.65°
- C.75°
- D.115°
Show answer
Answer: B: 65°
The three angles of a triangle add up to 180°. First add the two you know: 48 + 67 = 115. Then subtract from 180: 180 − 115 = 65°. Choice D, 115°, is the sum of the two given angles, which is a step on the way to the answer but not the answer itself.
Example 3
A circle has a radius of 5 inches. What is its circumference? Use 3.14 for pi.
- A.15.7 in
- B.31.4 in
- C.62.8 in
- D.78.5 in
Show answer
Answer: B: 31.4 in
Circumference is 2 × pi × radius, so 2 × 3.14 × 5 = 3.14 × 10 = 31.4 inches. Choice D, 78.5, is the area of the circle, since 3.14 × 5 × 5 = 78.5, and it is the classic trap of using the wrong circle formula. Choice A, 15.7, forgets the 2 in the formula.
Example 4
A rectangular box is 8 inches long, 5 inches wide, and 3 inches tall. What is its volume?
- A.16 in³
- B.40 in³
- C.120 in³
- D.158 in³
Show answer
Answer: C: 120 in³
Volume is length × width × height: 8 × 5 = 40, and 40 × 3 = 120 cubic inches. Choice B, 40, stops after multiplying only two of the three dimensions. Choice D, 158, is the total surface area of the box, which is a different measurement with a different unit.
Traps to watch for
- Radius versus diameter. If a question gives you a diameter of 10, the radius is 5. Using 10 in a circle formula doubles or quadruples your answer, and that wrong answer will be one of the choices.
- Forgetting the one half. Triangle area is half of base times height. A triangle answer equal to the full product of base and height is always wrong.
- Mixing up area and perimeter, or area and circumference. Perimeter and circumference measure distance around, in plain units. Area measures the space inside, in square units. Read the question word carefully.
- Ignoring units. If a rectangle is 2 feet by 18 inches, convert to the same unit before you multiply. The answer choices will include the unconverted product.
Pacing
Geometry questions can take a little longer than pure arithmetic because there is a formula to recall and a sketch to draw, but you still have only about 42 seconds per question on average. Grab the ones with a formula you know cold, and if you are stuck on an angle puzzle, pick your best choice and move on since there is no guessing penalty. Make sure every question has an answer before time is called.
Practice
Find out which geometry formulas you actually remember under a clock with our free diagnostic.
Shaky on the skill itself? Start with the lessons on geometry and measurement and units, then come back to the question format.