Math: Geometry and Measurement
Angles, triangles, area, circles, surface area and volume: the Grade 7 geometry that lands on every SHSAT form, with no formula sheet to lean on.
25 topics · 102 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Mathematical Terms and Symbols You Must Know
- How it worksHow do the marks for parallel, perpendicular, congruent sides and equal angles get read off a drawn figure?
- Common mistakeWhy isn't every pair of lines that never cross on the page a pair of parallel lines?
- Worked exampleFind both angles when perpendicular lines meet and a ray splits one right angle into 2x and 3x degrees.
- Going furtherWhat changes in a problem when it says two segments are congruent instead of saying they are equal?
Your turnmedium1 of 3
Two perpendicular lines meet, and a ray splits one of the right angles into parts measuring 2x degrees and 6x degrees. Find the measure of the larger part, in degrees.Complementary, Supplementary and Adjacent Angles
- How it worksHow do complementary and supplementary angles differ once you draw both pairs side by side?
- Worked exampleFind the supplement of an angle whose complement measures 23 degrees.
- Common mistakeWhy doesn't a pair of angles have to be touching before you can call them supplementary?
- Going furtherWhat happens to the supplement of an angle as that angle grows from 10 degrees toward 170 degrees?
Your turnmedium1 of 3
An angle has a complement measuring 59 degrees. Find the measure of that angle's supplement, in degrees.Vertical Angles and Linear Pairs
- How it worksHow does the straight-line pair at a crossroads force the opposite angles to be equal?
- Worked exampleFind x when two crossing lines make vertical angles of 4x plus 10 and 6x minus 18 degrees.
- Common mistakeWhy isn't a pair of opposite angles at a crossing supplementary unless both of them are right angles?
Your turneasy1 of 3
Two crossing lines make vertical angles measuring 4x plus 10 degrees and 6x minus b degrees. Write an expression for x, in terms of b.Angles Formed When a Transversal Cuts Parallel Lines
- How it worksHow do corresponding, alternate interior and alternate exterior angles get their names from where they sit?
- How it worksWalk me through why two angles on the same side of a transversal between parallel lines add to 180 degrees.
- Worked exampleFind all eight angles when a transversal cuts two parallel lines and one of them measures 118 degrees.
- Common mistakeWhy doesn't a pair of equal alternate interior angles prove anything when the two lines are not parallel?
- Going furtherWhat happens to the eight angles when the two lines slowly tilt out of being parallel?
Your turneasy1 of 3
A transversal cuts two parallel lines, and one of the eight angles measures 62 degrees, which is acute. Find the measure of the largest of the eight angles, in degrees.Writing an Equation for an Unknown Angle
- How it worksHow do you turn words about a figure into one equation with the unknown angle on a single side?
- Worked exampleFind x when an angle of 3x plus 10 degrees and an angle of x plus 4 degrees are complementary.
- Worked exampleFind the two pieces when a straight angle is split into parts of 5x minus 7 and 2x plus 12 degrees.
- Common mistakeWhy doesn't adding up every angle label in a figure give you a usable equation on its own?
Your turneasy1 of 3
An angle of 6x plus c degrees and an angle of x plus 4 degrees are complementary. Write an expression for x, in terms of c.The Triangle Angle Sum
- How it worksHow does tearing the three corners off a paper triangle and lining them up prove they total 180 degrees?
- How it worksWalk me through the parallel-line argument for why a triangle's three angles must total 180 degrees.
- Worked exampleFind the third angle of a triangle whose other two measure 47 and 68 degrees, then name its type.
- Common mistakeWhy isn't there a triangle with two angles of 95 degrees, no matter how long you make the sides?
- Going furtherWhat happens to the angle total when you add a fourth corner and make the shape a quadrilateral?
Your turneasy1 of 3
A triangle has angles measuring 47 degrees and 93 degrees. Find the third angle, in degrees.Exterior Angles of a Triangle
- How it worksHow does the straight-line pair at one vertex turn into the exterior angle rule?
- Worked exampleFind the exterior angle at the third vertex of a triangle whose other two angles are 52 and 61 degrees.
- Going furtherWhat happens when you add up all three exterior angles of any triangle at once?
Your turneasy1 of 3
A triangle has two angles measuring 52 degrees and 86 degrees. Find the exterior angle at the third vertex, in degrees.Classifying Triangles by Sides and Angles
- How it worksHow do the side names and the angle names split every triangle into a small grid of types?
- Common mistakeWhy isn't a triangle that looks like it has a square corner a right triangle until the numbers say so?
- Worked exampleFind both names for a triangle with angles of 30, 30 and 120 degrees.
- Going furtherWhat happens to a triangle's classification as one angle slides from 89 to 90 to 91 degrees?
Your turneasy1 of 3
An isosceles triangle has a vertex angle of 104 degrees. Find the measure of each base angle, in degrees.The Triangle Inequality and When a Triangle Exists
- How it worksHow does the longest stick decide whether the other two can reach across and meet?
- Worked exampleFind every whole-number length the third side can take when two sides of a triangle are 7 and 12.
- Common mistakeWhy isn't a list of three angles enough to pin down one single triangle the way three sides does?
- Going furtherWhat happens to the shape when two short sides add to exactly the length of the long one?
Your turneasy1 of 3
Two sides of a triangle measure 7 units and 24 units. What is the largest whole-number length the third side can take?Perimeter of Rectangles, Triangles and Odd Polygons
- How it worksHow do you find the perimeter of an L-shaped room when two of the side lengths were never labelled?
- Worked exampleFind the perimeter of a rectangle whose length is 3 times its width and whose area is 108 square cm.
- Worked exampleFind the sixth side of a hexagon with a perimeter of 48 whose other sides are 6, 9, 4, 11 and 7.
Your turnhard1 of 3
A rectangle's length is 5 times its width, and its width is w centimetres. Write an expression for its perimeter, in centimetres, in terms of w.Area of Rectangles and Parallelograms
- How it worksHow does cutting a triangle off one end of a parallelogram and moving it build a rectangle?
- Common mistakeWhy doesn't the slanted side of a parallelogram ever get used as the height in the area formula?
- Worked exampleFind the area of a parallelogram with a base of 14 cm, a slanted side of 10 cm and a height of 8 cm.
- Going furtherWhat happens to the area of a rectangle when you double only its width and leave its length alone?
Your turneasy1 of 3
A parallelogram has a base of 17 cm, a slanted side of 12 cm and a height of h cm. Write an expression for its area, in square centimetres, in terms of h.Area of Triangles and Trapezoids
- How it worksHow do two copies of the same trapezoid snap together into a single parallelogram?
- How it worksHow do you pick the right height for a triangle when the height lands outside the triangle itself?
- Worked exampleFind the area of a trapezoid with parallel sides of 9 and 15 inches and a height of 6 inches.
- Worked exampleFind the height of a triangle whose area is 84 square feet and whose base measures 24 feet.
- Common mistakeWhy isn't the height of a triangle just the length of the side that slants up from the base?
Your turneasy1 of 3
A trapezoid has parallel sides of 12 inches and b inches and a height of 8 inches. Write an expression for its area, in square inches, in terms of b.Composite Figures and Shaded Regions
- How it worksHow do you decide whether to cut a composite figure into pieces or subtract a hole from a big rectangle?
- Worked exampleFind the area of a track made from a 40 by 20 meter rectangle with a half circle stuck on each end.
- Worked exampleFind the shaded region left when a circle of radius 5 is cut out of a square with sides of 10.
- Common mistakeWhy doesn't adding the perimeters of the pieces give the perimeter of the whole composite shape?
- Going furtherWhat happens to a shaded-region answer when the hole is slid to a new spot but never resized?
Your turnmedium1 of 3
A running track encloses a 51 metre by w metre rectangle with a half circle stuck on each of the two short ends, each half circle having diameter w. Write an expression for the total enclosed area, in square metres, in terms of w.Radius, Diameter and Circumference
- How it worksHow does unrolling a circle's rim into a straight segment show where 2 times π times r comes from?
- How it worksHow do radius, diameter and circumference each respond when a circle is scaled up?
- Worked exampleFind the circumference of a bicycle wheel of diameter 26 inches, then how far 100 turns carry it.
- Common mistakeWhy isn't π equal to 22 over 7, even though that fraction keeps getting used in class?
- Going furtherWhat happens to the circumference of a circle when you double its radius?
Your turneasy1 of 3
A bicycle wheel has a diameter of d inches. Write an expression, in terms of d, for how far it carries the bicycle in 154 complete turns, leaving π in your answer.Area of a Circle
- How it worksHow do thin pie slices rearranged into a near-rectangle show that area is π times r squared?
- Worked exampleFind the area of a circular garden whose circumference is 44 meters, using 22 over 7 for π.
- Common mistakeWhy doesn't doubling the radius of a pizza only double the amount of pizza you get?
- Going furtherWhat happens to the area of a circle when its diameter is cut to one third of the old size?
Your turnhard1 of 3
A circular garden has a circumference of 68 metres. Find its area in square metres, using 22/7 for π.Answers Left in Terms of π
- How it worksHow do you carry π as a symbol all the way through a multi-step circle problem?
- Worked exampleFind the area and the circumference of a circle of radius 9, both left in terms of π.
- Common mistakeWhy isn't 3.14 a safe swap for π when the answer options are written with π still in them?
Your turneasy1 of 3
Write an expression for the combined area of 7 circles each of radius r units, in terms of r, leaving π in your answer.Nets and the Surface Area of Prisms and Pyramids
- How it worksHow does the net of a rectangular prism split into three matching pairs of rectangles?
- How it worksHow do the four triangles in a square pyramid's net use the slant height instead of the pyramid's height?
- Worked exampleFind the surface area of a rectangular prism measuring 8 by 5 by 3 centimeters.
- Worked exampleFind the surface area of a square pyramid with a base edge of 10 m and a slant height of 13 m.
- Going furtherWhat happens to the surface area of a cube when every one of its edges is doubled?
Your turneasy1 of 3
Write an expression for the surface area of a rectangular prism measuring 8 by 7 by c centimetres, in square centimetres, in terms of c.Volume of Right Prisms and Cubes
- How it worksHow does base area times height cover every single right prism with one formula?
- Worked exampleFind the volume of a triangular prism whose base has legs of 6 and 8 and whose length is 15.
- Going furtherWhat happens to a cube's volume when its edge is tripled, and why is the answer not 3 times bigger?
- Common mistakeWhy isn't volume ever measured in the same units as the area of one of its faces?
Your turneasy1 of 3
A triangular prism has a right-triangular base with legs of 8 units and b units, and a length of 16 units. Write an expression for its volume, in cubic units, in terms of b.Volume and Surface Area of Cylinders
- How it worksHow does base area times height still work for a cylinder when the base is a circle, not a polygon?
- Worked exampleFind the volume of a cylinder with a radius of 4 cm and a height of 10 cm, left in terms of π.
- Worked exampleFind the surface area of a closed can of radius 3 m and height 7 m by unrolling its label flat.
- Common mistakeWhy doesn't the label wrapped around a can become a circle when you peel it off and flatten it?
Your turneasy1 of 3
A cylinder has a radius of r cm and a height of 15 cm. Write an expression for its volume, in cubic centimetres, in terms of r, leaving π in your answer.Composite Solids
- How it worksHow do you work out which faces disappear when two solids are glued together?
- Worked exampleFind the volume of a barn made from a 12 by 6 by 4 m prism topped by a triangular prism 3 m tall.
- Going furtherWhat happens to the total surface area of two blocks when you push them together face to face?
Your turnmedium1 of 3
A barn is made from a rectangular prism 14 m long, 8 m wide and 6 m high, topped by a triangular prism roof h m tall running the full 14 m length. Write an expression for the total volume, in cubic metres, in terms of h.Scale Drawings and Redrawing at a New Scale
- How it worksHow does a scale factor turn every length in a drawing into a real length with one multiplication?
- Worked exampleFind the real area of a yard drawn 6 cm by 4 cm on a plan where 1 cm stands for 3 meters.
- Worked exampleFind the new side lengths when a 12 by 8 drawing is redrawn at a scale that turns the 12 into 18.
- Common mistakeWhy doesn't doubling every length on a map only double the ground it covers?
- Going furtherWhat happens to the areas in a scale drawing when the scale changes from 1 to 50 into 1 to 100?
Your turnmedium1 of 3
A yard is drawn 8 cm by 6 cm on a plan where 1 cm stands for k metres. Write an expression for the real area of the yard, in square metres, in terms of k.Cross Sections of Three-Dimensional Solids
- How it worksHow do slices parallel to the base and slices perpendicular to it differ for a rectangular prism?
- Worked exampleFind the shape and the measurements of a horizontal slice through a 6 by 4 by 10 rectangular prism.
- Common mistakeWhy isn't a slice through a square pyramid always a square?
- Going furtherWhat happens to the cross section of a square pyramid as the slicing plane rises toward the tip?
Your turneasy1 of 3
A rectangular prism measures 6 by 9 by 10 units, standing on its 6 by 9 base. A horizontal slice is taken. Find the area of that slice, in square units.Congruence and Corresponding Parts
- How it worksHow do you read corresponding sides and angles out of the letter order in two triangle names?
- Worked exampleFind the missing side and angle when triangle ABC matches triangle DEF, AB is 7 and angle E is 52.
- Common mistakeWhy doesn't matching area and matching perimeter make two figures congruent?
Your turnmedium1 of 3
Triangle ABC is congruent to triangle DEF. Side AB measures 7 units, side BC measures 13 units, and the perimeter of DEF is 30 units. Find the length of CA, in units.Similar Figures and Finding a Missing Side
- How it worksHow does a single scale factor connect every pair of matching sides in two similar figures?
- How it worksHow do two equal angles on their own force a pair of triangles to be similar?
- Worked exampleFind the missing side when a triangle of sides 6, 8 and 10 is similar to one whose shortest side is 15.
- Worked exampleFind the height of a flagpole casting a 24 m shadow while a 1.8 m person casts a 3 m shadow.
- Going furtherWhat happens to area and to perimeter when a whole figure is scaled up by a factor of 4?
Your turneasy1 of 3
A triangle with sides 6, 9 and 12 units is similar to a triangle whose shortest side is s units. Write an expression for the longest side of the second triangle, in terms of s.Every Formula You Have to Memorize
- How it worksHow do the area rules for rectangle, triangle, parallelogram, trapezoid and circle all grow from one idea?
- Worked exampleFind the volume and the surface area of a cube of edge 7 using only formulas recalled from memory.
- Going furtherWhat happens to your time budget when you have to rebuild a forgotten formula mid-question?
- Ties togetherHow do the perimeter, area and volume rules decide which units the final answer has to wear?
Your turneasy1 of 3
From memory, with no formula sheet available, write an expression for the combined volume of 7 cubes each of edge e units, in cubic units, in terms of e.