Grade 8 Geometry Worksheets — Pythagoras and Square Roots
Eighth grade geometry here is one idea taught in four steps, and the four sheets are meant to be printed in order: Square Roots and Cube Roots, Pythagoras: Find the Hypotenuse, Pythagoras: Find a Missing Side, and Distance Between Two Points. Each one ends where the next one begins, so a student who finishes the last sheet has used every step of the first.
Square Roots and Cube Roots comes first because everything after it finishes with a root. It opens on Squares and square roots, moves to Cubes and cube roots, and closes with Two roots in one question, where a square root and a cube root sit in the same line and have to be kept apart. Pick a sheet, choose a level inside it, set the question count, tick the answer key box, and print — and the link in your address bar rebuilds the identical sheet whenever you want it again.
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Geometry · Grade 8
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Grade 8 · 3 pages · A4More Grade 8 geometry skills
Each one generates a different kind of question, with its own three levels and answer key.
- Square Roots and Cube RootsFind the number that was squared or cubed, then combine two roots in one question.
- Pythagoras: Find the HypotenuseBoth shorter sides are given — square them, add, and find the square root.
- Pythagoras: Find a Missing SideThe hypotenuse and one shorter side are given — subtract before you take the root.
- Distance Between Two PointsCount two sides off a grid, in one quadrant or all four, and find the slanting line between the points.
What each level generates
Grade 8 square roots and cube roots — the exact skill behind each difficulty setting, straight from the generator.
- Level 1Squares and square roots
- Level 2Cubes and cube roots
- Level 3Two roots in one question
Grade levels follow the progression common to most English-language curricula — not one specific national curriculum. Every level shows exactly what it generates, so you can check the fit with your own program. See the full scope & sequence →
Why square roots are on a geometry page
Square roots are arithmetic, not geometry, and they are here for one reason: they are the last step of every other sheet in this set. A student who can set up 9² + 40² = 1681 and then stalls has not failed at Pythagoras, they have run out of square roots — and sending them to a different topic to fix that breaks the lesson in half. Keeping the drill on the same page means the prerequisite is one click away from the thing it is a prerequisite for.
If your student already knows their squares and cubes cold, skip the sheet entirely and start with Pythagoras: Find the Hypotenuse. If they do not, twenty questions of squares takes five minutes and saves twenty. Worth knowing by heart: every square up to 20 × 20, and the eight cube numbers under a thousand — 8, 27, 64, 125, 216, 343, 512 and 729. Those eight are the only cube roots any of these sheets will ever ask for.
The two directions of Pythagoras are two different sheets
Find the Hypotenuse and Find a Missing Side use one rule, and students who have only met it going forwards reliably get the backwards version wrong in the same way: they add when the question wanted a subtraction. Given a hypotenuse of 17 cm and a side of 15 cm, adding the squares gives 514 and a nonsense answer; subtracting gives 64 and a side of 8 cm. That is why the two are separate pages here rather than two levels of one — the mistake is common enough to deserve a whole sheet of practice, not a third of one.
There is a check that catches it every time, and it is worth insisting on: a missing shorter side must always come out SMALLER than the hypotenuse. Ask for that comparison before the answer is written down. Distance Between Two Points is the same rule again with the two shorter sides read off a grid — across gap, up gap, square, add, root. Two of its three levels now PRINT that grid and name only the letters, so the coordinates are read off the picture rather than taken from the sentence: level 1 draws the slanting line itself, and level 3 plots the two points and leaves the triangle to be seen. Level 3 puts them in opposite quadrants, where the gap between 2 and −6 is 8 and not 4 — counting straight through zero rather than stopping at it is the whole level.
Level 2 prints no grid, and that is a decision rather than an omission: its points are far enough apart that a grid a student could actually count would not fit on the paper, so the coordinates are stated and the gaps are subtracted instead of counted. It is the level to reach for once counting squares has become a crutch.
Frequently asked questions
▸What is the Pythagorean theorem?
In a right-angled triangle, the two shorter sides squared and added give the longest side squared: a² + b² = c². The longest side, opposite the right angle, is called the hypotenuse. Find the Hypotenuse practises the rule going forwards, and Find a Missing Side runs it backwards as a subtraction.
▸Are the answers always whole numbers?
Yes. Every triangle on these sheets is built from a Pythagorean triple — 3-4-5, 5-12-13, 8-15-17, 7-24-25, 20-21-29, 9-40-41 or a multiple of one — so every hypotenuse and every missing side is an exact whole number. Nothing is rounded, which means a wrong answer always points to a wrong method rather than to a decimal that slipped.
▸How do you find the distance between two points?
Subtract the two x values for the horizontal gap and the two y values for the vertical gap, ignoring any minus signs, then treat those as the shorter sides of a right-angled triangle. From (1, 5) to (5, 8) the gaps are 4 and 3, so the distance is 5 units. Distance Between Two Points draws the grid on two of its three levels, so the gaps are counted off the picture rather than read out of the sentence, and it builds up to a pair placed in opposite quadrants.
▸Why is there no Grade 7 geometry page?
Because there is nothing honest to put on it. Circles are the usual Grade 7 topic and they already have two sheets on the Grade 6 page, where the UK curriculum allows them; the rest of Grade 7 geometry is scale drawings and cross-sections, which need a diagram a length is measured off with a ruler. These sheets do print pictures — coordinate grids, plans of rectangles and triangles, circles, a cuboid, Grade 4's symmetry patterns and the angle cards at Grades 4 to 6 — but every one of them is read by counting, by looking, or off a number already written on the drawing, never by measuring, and a scale drawing is the opposite of that. An empty year is more useful than a padded one — print the Grade 6 circle sheets in Year 7 and come back here.
▸Do these worksheets need a diagram or a calculator?
No calculator — the numbers are chosen so the squaring and the roots can be done on paper. A diagram is printed where the question is about one: Distance Between Two Points draws the grid on two of its three levels, and the gaps are counted off it rather than read out of the sentence. The Pythagoras sheets draw the right-angled triangle on their first level, with the two sides you are given marked on it and the side you are asked for left blank; their second and third levels are written in words, because the triangles that level 2 works with are too large to print to scale. Tick the answer key box and the answers print on their own page at the end.