Why it matters on the HSPT
Geometry is about a fifth of the Mathematics section and the whole of the geometric comparison items in Quantitative Skills. The test does not ask for proofs or constructions. It asks you to know a small set of formulas and facts and apply them to a labeled figure, usually in one or two steps. The figures are drawn to help, but the numbers on them are what count.
The rules
Angles. A full turn is 360 degrees, a straight line is 180, and a right angle, marked with a small square, is 90. Two angles that make a straight line are supplementary and add to 180. Two that make a right angle are complementary and add to 90. When two lines cross, the angles opposite each other, called vertical angles, are equal.
Parallel lines and a transversal. Parallel lines never meet; on a figure they carry matching arrowheads. A line that cuts across two parallel lines is a transversal, and it makes eight angles that come in only two sizes. Every acute angle equals every other acute angle, every obtuse angle equals every other obtuse angle, and an acute plus an obtuse makes 180. So if one angle is 62, every other angle at both crossings is 62 or 118. A single known angle unlocks all eight.
Triangles. The three angles add to 180. An equilateral triangle has three equal sides and three 60-degree angles. An isosceles triangle has two equal sides, and the angles opposite those sides are equal. A right triangle has one 90-degree angle, so its other two add to 90. The largest angle is always opposite the longest side.
Right triangles. The Pythagorean theorem says the two shorter sides squared and added equal the longest side squared. The test relies on 3-4-5 and its multiples: 6-8-10, 9-12-15, and 5-12-13 occasionally. If two sides match one of these, the third is known without any arithmetic.
Similar triangles. Two triangles are similar when they have the same angles, which makes them the same shape at different sizes. Their sides are in proportion: every side of the bigger one is the same multiple of the matching side of the smaller one. If a triangle with sides 3, 4, and 5 is similar to one whose shortest side is 9, the multiplier is 3 and the other sides are 12 and 15. Find the multiplier from the one pair of matching sides you know, then apply it. Congruent triangles are the special case where the multiplier is 1: same shape, same size, with tick marks showing which sides match.
Quadrilaterals. The angles add to 360. A rectangle has four right angles and opposite sides equal; a square is a rectangle with all sides equal. A parallelogram has opposite sides parallel and equal.
Kinds of quadrilaterals. A parallelogram has both pairs of opposite sides parallel. A rhombus is a parallelogram with all four sides equal but no right angles required. A trapezoid has exactly one pair of parallel sides. Every square is a rectangle and a rhombus, and every rectangle and rhombus is a parallelogram, but a trapezoid is none of these. When a question asks which shape is not a parallelogram, the trapezoid, or any shape with a curve, is the answer.
Polygons and their angles. A polygon is named by its number of sides: triangle 3, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7, octagon 8. To find the total of the interior angles, cut the shape into triangles from one corner. A shape with n sides makes n minus 2 triangles, so its angles add to n minus 2, times 180. A pentagon has 3 times 180, which is 540; a hexagon has 720. A regular polygon has equal angles, so divide: each angle of a regular hexagon is 720 divided by 6, which is 120.
Perimeter. Add the sides. For a rectangle, that is 2 times the length plus 2 times the width. For a square, 4 times a side. Perimeter is in the same unit as the sides.
Area. A rectangle is length times width, a square is side squared, a triangle is one half of base times height, and a parallelogram is base times height. The height is always perpendicular to the base, not the slanted side. Area is in square units. For an odd shape, split it into rectangles and add, or subtract a cut-out piece from a bigger rectangle.
Composite shapes. For an L or T shape, find every missing side before computing anything. Opposite sides of the outline add up: if the full height on the left is 5 and a piece on the right is 1, the other piece on the right is 4. Perimeter is then the sum of all the sides, including the ones you found. Area is the sum of the rectangles you cut the shape into, or a big rectangle minus the notch. Draw the cut line on the figure; it prevents double counting.
Circles. The radius is the distance from the center to the edge, and the diameter is twice the radius. Circumference is pi times the diameter. Area is pi times the radius squared. Use 3.14 or 22 over 7 for pi, or leave the answer in terms of pi when the choices do.
Chords. A chord is any segment with both endpoints on the circle. A diameter is a chord that passes through the center, and it is the longest chord. A radius is not a chord, because one of its ends is the center. On a figure with several segments drawn, the chord is the one that touches the circle at both ends.
Volume. A box is length times width times height, in cubic units. A cube is the edge cubed.
Surface area. The surface area of a box is the total area of its six faces. Opposite faces match, so find the three different faces and double each: 2 times length times width, plus 2 times length times height, plus 2 times width times height. A 9 by 3 by 5 box has faces of 27, 45, and 15, so its surface area is 2 times 87, which is 174 square units. A cube has six identical faces, so its surface area is 6 times the edge squared. Surface area is in square units even though the object is solid.
Cylinders. A soup can is a cylinder. Its volume is the area of the circular base times the height: pi times the radius squared times the height. The test leaves pi in the answer or tells you to use 3.14. A basketball is a sphere and a cereal box is a rectangular prism; the test sometimes just asks you to name the solid.
Coordinates. A point is written as x then y, across then up. The distance between two points on the same horizontal or vertical line is the difference of their coordinates. A rectangle drawn on the grid has its sides along the lines, so its dimensions are read straight off the coordinates.
Worked examples
Example 1
Two angles of a triangle measure 48° and 67°. What is the third angle?
- A.55°
- B.65°
- C.75°
- D.115°
Show answer
Answer: B: 65°
The three angles add to 180. Forty-eight plus 67 is 115, and 180 minus 115 is 65. Choice D, 115, is the sum of the two given angles, not the missing one.
Example 2
A rectangle is 9 cm long and 4 cm wide. What is its perimeter?
- A.13 cm
- B.26 cm
- C.36 cm
- D.36 cm²
Show answer
Answer: B: 26 cm
Perimeter is 2 times 9 plus 2 times 4, which is 18 plus 8, which is 26 cm. Choice C, 36, is the area, and choice D is the area with the right unit for area but the wrong quantity.
Example 3
What is the area of a triangle with a base of 10 inches and a height of 6 inches?
- A.16 sq in
- B.30 sq in
- C.60 sq in
- D.120 sq in
Show answer
Answer: B: 30 sq in
Area is one half of base times height: one half of 10 times 6, which is one half of 60, which is 30 square inches. Choice C forgets the one half, which is the most common triangle error on the test.
Example 4
A circle has a radius of 7 cm. Using π ≈ 22/7, what is its circumference?
- A.22 cm
- B.44 cm
- C.154 cm
- D.308 cm
Show answer
Answer: B: 44 cm
Circumference is pi times diameter. The diameter is 14, and 22 over 7 times 14 is 22 times 2, which is 44 cm. Choice C, 154, is the area, pi times 49.
Example 5
A right triangle has legs of 6 and 8. What is the length of the longest side?
- A.7
- B.10
- C.14
- D.100
Show answer
Answer: B: 10
Six and 8 are the 3-4-5 triangle doubled, so the longest side is 5 doubled, 10. By the theorem, 36 plus 64 is 100, and the root of 100 is 10. Choice D stops at 100 without taking the root.
Example 6
A box is 5 inches long, 4 inches wide, and 3 inches tall. What is its volume?
- A.12 cu in
- B.47 cu in
- C.60 cu in
- D.94 cu in
Show answer
Answer: C: 60 cu in
Volume is length times width times height: 5 times 4 times 3, which is 60 cubic inches. Choice A adds the three dimensions, and choice D is the surface area.
Example 7
Two lines cross so that one of the angles formed measures 35°. What is the measure of the angle directly across from it?
- A.35°
- B.55°
- C.145°
- D.325°
Show answer
Answer: A: 35°
Angles across from each other at a crossing are vertical angles, and vertical angles are equal, so the answer is 35 degrees. Choice C, 145, is the angle next to it, which is supplementary.
Example 8
Triangle ABC is similar to triangle DEF. Side AB is 6, side BC is 9, and the matching side DE is 24. What is the length of EF?
- A.27
- B.36
- C.21
- D.16
Show answer
Answer: B: 36
Match the sides first: AB matches DE, so the multiplier from the small triangle to the big one is 24 divided by 6, which is 4. BC matches EF, so EF is 9 times 4, which is 36. Choice C, 21, adds the difference of 18 to 9 instead of multiplying; similar triangles scale, they do not grow by a fixed amount.
Example 9
An L-shaped floor has a top edge of 4 meters, a right edge of 5 meters, a bottom edge of 7 meters, and a short left edge of 1 meter, with two sides not labeled. What is the perimeter?
- A.17 m
- B.24 m
- C.27 m
- D.35 m
Show answer
Answer: B: 24 m
Find the missing sides. The full height on the right is 5 and the short left edge is 1, so the other vertical piece is 4. The full width along the bottom is 7 and the top is 4, so the missing horizontal piece is 3. Now add all six sides: 4 plus 5 plus 7 plus 1 plus 4 plus 3 is 24. Choice A adds only the four labeled sides. Choice C, 27, is the area in square meters.
Example 10
A box is 8 inches long, 4 inches wide, and 2 inches tall. What is its surface area?
- A.14 sq in
- B.64 sq in
- C.112 sq in
- D.128 sq in
Show answer
Answer: C: 112 sq in
The three different faces are 8 by 4, 8 by 2, and 4 by 2, with areas 32, 16, and 8. Each appears twice, so the surface area is 2 times 56, which is 112 square inches. Choice B, 64, is the volume. Choice D doubles the volume, and choice A is the sum of the three dimensions.
Common mistakes
- Forgetting the one half in the triangle area. Base times height is a rectangle, not a triangle.
- Confusing perimeter with area. Perimeter is a distance around; area covers the inside and comes in square units.
- Using the radius where the diameter belongs. Circumference uses the diameter; area uses the radius squared.
- Adding dimensions for volume. Volume multiplies all three.
- Using the slanted side as the height. Height is perpendicular to the base.
Adding instead of scaling on similar triangles. If one side went from 6 to 24, every side is multiplied by 4. Nothing is added.
Skipping the missing sides on an L shape. The unlabeled sides count toward the perimeter. Work them out from the sides across from them before you add.
Confusing surface area with volume. Volume multiplies the three dimensions once. Surface area adds up six faces. If your surface-area answer equals length times width times height, you found the volume.
Practice
The two newest topics, the coordinate plane, sits alongside these rules. These formulas are the substance of basic geometry items and geometric comparison items, and shapes appear inside word problems about fencing, tiling, and painting. The measurement lesson covers the square and cubic unit conversions.