ClipArt ETC
Plan, front, and side views of a square pyramid after revolving pyramid through 30° with horizontal plane.

Square Pyramid Revolved 30°

Plan, front, and side views of a square pyramid after revolving pyramid through 30° with horizontal…

Illustration of the intersection of a square pyramid and a plane.

Intersection of Square Pyramid and a Plane

Illustration of the intersection of a square pyramid and a plane.

Illustration of the intersection of a square pyramid and a plane.

Intersection of Square Pyramid and a Plane

Illustration of the intersection of a square pyramid and a plane.

Illustration of an irregular convex polygon with 16 sides that has symmetry. It could be used to show rotation of a square.

Irregular Convex Polygon

Illustration of an irregular convex polygon with 16 sides that has symmetry. It could be used to show…

Illustration of a square bifrustum created by three parallel planes of squares with the middle plane largest. The top and bottom squares are congruent. It can be the combination of two congruent frustums across a plane of symmetry, or as a bipyramid with truncated vertices.

Square Bifrustum

Illustration of a square bifrustum created by three parallel planes of squares with the middle plane…

Illustration of a rectangular bipyramid viewed at an angle. A bipyramid, or dipyramid, is formed by joining two congruent pyramids at their bases.

Rectangular Bipyramid

Illustration of a rectangular bipyramid viewed at an angle. A bipyramid, or dipyramid, is formed by…

Illustration of a rectangular bipyramid. A bipyramid, or dipyramid, is formed by joining two congruent pyramids at their bases.

Rectangular Bipyramid

Illustration of a rectangular bipyramid. A bipyramid, or dipyramid, is formed by joining two congruent…

Illustration of a rectangular bipyramid. A bipyramid, or dipyramid, is formed by joining two congruent pyramids at their bases.

Rectangular Bipyramid

Illustration of a rectangular bipyramid. A bipyramid, or dipyramid, is formed by joining two congruent…

Illustration of 28 congruent cubes placed in the shape of a square. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

28 Congruent Cubes Placed in the Shape of a Square

Illustration of 28 congruent cubes placed in the shape of a square. A 3-dimensional representation on…

Illustration of 36 congruent cubes stacked at various heights with outer edges forming a square. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

36 Stacked Congruent Cubes

Illustration of 36 congruent cubes stacked at various heights with outer edges forming a square. A 3-dimensional…

Illustration of 56 congruent cubes stacked in twos in the shape of a square. A 3-dimensional representation on a 2-dimensional surface that can be used for testing depth perception and identifying and counting cubes, edges, and faces.

56 Stacked Congruent Cubes

Illustration of 56 congruent cubes stacked in twos in the shape of a square. A 3-dimensional representation…

Illustration of a square circumscribed about a regular dodecagon. This could also be described as a dodecagon inscribed in a square.

Square Circumscribed About A Dodecagon

Illustration of a square circumscribed about a regular dodecagon. This could also be described as a…

Illustration of 2 squares; one inscribed in a regular dodecagon and the other circumscribed about the same dodecagon.

Squares Inscribed and Circumscribed About a Regular Dodecagon

Illustration of 2 squares; one inscribed in a regular dodecagon and the other circumscribed about the…

Illustration of a square inscribed in an regular dodecagon. This could also be described as a regular dodecagon circumscribed about a square.

Square Inscribed In A Dodecagon

Illustration of a square inscribed in an regular dodecagon. This could also be described as a regular…

Illustration of 3 squares inscribed in an regular dodecagon. Each vertex of the dodecagon is also a vertex of one of the squares.

3 Square Inscribed In A Dodecagon

Illustration of 3 squares inscribed in an regular dodecagon. Each vertex of the dodecagon is also a…

Illustration of a square inscribed in a square. The interior square is rotated 45° in relation to the exterior square.

Square Inscribed In A Square

Illustration of a square inscribed in a square. The interior square is rotated 45° in relation to…

Illustration of a square inscribed in a square that is inscribed in another square. Each successive square is rotated 45° in relation to the previous square.

Squares Inscribed In Squares

Illustration of a square inscribed in a square that is inscribed in another square. Each successive…

Illustration of a square inscribed in a square that is inscribed in another square. Each successive square is rotated 45° in relation to the previous square. Diagonals of the largest square are shown.

Squares Inscribed In Squares

Illustration of a square inscribed in a square that is inscribed in another square. Each successive…

Illustration of a square inscribed in a square that is inscribed in another square. Each successive square is rotated 45° in relation to the previous square. Line segments are drawn connecting the vertices of the smallest to the vertices of the largest square.

Squares Inscribed In Squares

Illustration of a square inscribed in a square that is inscribed in another square. Each successive…

Illustration of 2 concentric squares whose vertices are connected by line segments.

2 Concentric Squares

Illustration of 2 concentric squares whose vertices are connected by line segments.

Illustration of 2 concentric squares.

2 Concentric Squares

Illustration of 2 concentric squares.

Illustration of 2 concentric squares.

2 Concentric Squares

Illustration of 2 concentric squares.

Illustration of 3 concentric squares that are equally spaced.

3 Concentric Squares

Illustration of 3 concentric squares that are equally spaced.

Illustration of 4 concentric squares that are equally spaced.

4 Concentric Squares

Illustration of 4 concentric squares that are equally spaced.

Illustration of 2 congruent squares that have the same center. One square has been rotated 45° in relation to the other.

2 Congruent Rotated Squares

Illustration of 2 congruent squares that have the same center. One square has been rotated 45° in…

Illustration of 4 congruent squares that have the same center. Each square has been rotated 22.5° in relation to the one next to it.

4 Congruent Rotated Squares

Illustration of 4 congruent squares that have the same center. Each square has been rotated 22.5°…

Illustration of 8 congruent squares that have the same center. Each square has been rotated 11.25° in relation to the one next to it.

8 Congruent Rotated Squares

Illustration of 8 congruent squares that have the same center. Each square has been rotated 11.25°…

Illustration of 16 congruent squares that have the same center. Each square has been rotated 5.625° in relation to the one next to it.

16 Congruent Rotated Squares

Illustration of 16 congruent squares that have the same center. Each square has been rotated 5.625°…

Illustration of 3 congruent squares that have the same center. Each square has been rotated 30° in relation to the one next to it.

3 Congruent Rotated Squares

Illustration of 3 congruent squares that have the same center. Each square has been rotated 30°…

Illustration of 6 congruent squares that have the same center. Each square has been rotated 15° in relation to the one next to it.

6 Congruent Rotated Squares

Illustration of 6 congruent squares that have the same center. Each square has been rotated 15°…

Illustration of 12 congruent squares that have the same center. Each square has been rotated 7.5° in relation to the one next to it.

12 Congruent Rotated Squares

Illustration of 12 congruent squares that have the same center. Each square has been rotated 7.5°…

Illustration of a regular octagon circumscribed about a square. This could also be described as a square inscribed in a regular octagon.

Octagon Circumscribed About a Square

Illustration of a regular octagon circumscribed about a square. This could also be described as a square…

Illustration showing the construction of a golden rectangle. Beginning with a unit square, a line is then drawn from the midpoint of one side of the square to its opposite corner. Using that line, an arc is drawn that defines the length of the rectangle. Two quantities are considered to be in the golden ratio if (a+ b)/a = a/b which is represented by the Greek letter phi.

Construction Of A Golden Rectangle

Illustration showing the construction of a golden rectangle. Beginning with a unit square, a line is…

Illustration showing how the golden ratio in a regular pentagon (inscribed in a circle) can be found using Ptolemy's theorem. The lines that are bolded form a quadrilateral. Ptolemy's theorem says the square of b equals the sum of a squared and ab, which in turn gives the golden ratio.

Golden Ratio In A Pentagon

Illustration showing how the golden ratio in a regular pentagon (inscribed in a circle) can be found…

Illustration of a swimming pool and water hose that is in the shape of a hollow regular decagonal prism with regular decagons on the ends/bases and square faces.

Swimming Pool Shaped Like A Decagonal Prism

Illustration of a swimming pool and water hose that is in the shape of a hollow regular decagonal prism…

Illustration of a non-regular decagonal prism in the shape of a star. Then ends/bases are made of star-shaped decagons and the faces are squares.

Star-Shaped Decagonal Prism

Illustration of a non-regular decagonal prism in the shape of a star. Then ends/bases are made of star-shaped…

Side view of a non-regular decagonal prism in the shape of a star. Then ends/bases are made of star-shaped decagons and the faces are squares.

Star-Shaped Decagonal Prism

Side view of a non-regular decagonal prism in the shape of a star. Then ends/bases are made of star-shaped…

Illustration of a regular right heptagonal/septagonal prism with regular heptagons/septagons for bases and square faces. The hidden edges are shown.

Heptagonal/Septagonal Prism

Illustration of a regular right heptagonal/septagonal prism with regular heptagons/septagons for bases…

Illustration of a right hexagonal prism with hexagons for bases and square faces.

Hexagonal Prism

Illustration of a right hexagonal prism with hexagons for bases and square faces.

Illustration of a right hexagonal prism with hexagons for bases and square faces. The hidden edges are shown.

Hexagonal Prism

Illustration of a right hexagonal prism with hexagons for bases and square faces. The hidden edges are…

Illustration of 4 congruent rectangular prisms placed in the shape of a square. They are arranged to look like they are 3-dimensional rectangular solids coming out of the page.

4 Congruent Rectangular Prisms

Illustration of 4 congruent rectangular prisms placed in the shape of a square. They are arranged to…

Illustration of a square inscribed in a closed concave geometric figure with 24 sides in the shape of a 12-point star. The two figures are concentric.

Square Inscribed in a 12-Point Star

Illustration of a square inscribed in a closed concave geometric figure with 24 sides in the shape of…

Illustration of a square inscribed in a circle. This can also be described as a circle circumscribed about a square.

Square Inscribed In A Circle

Illustration of a square inscribed in a circle. This can also be described as a circle circumscribed…

Illustration of a square inscribed in a circle. This can also be described as a circle circumscribed about a square. The diagonal of the square is also the diameter of the circle.

Square Inscribed In A Circle

Illustration of a square inscribed in a circle. This can also be described as a circle circumscribed…

Illustration of a square, with diagonals drawn, inscribed in a circle. This can also be described as a circle circumscribed about a square. The diagonals, which are also the diameter of the circle, intersect at the center of both the square and the circle.

Square Inscribed In A Circle

Illustration of a square, with diagonals drawn, inscribed in a circle. This can also be described as…

Illustration of a square, with diagonals drawn, circumscribed about a circle. This can also be described as a circle inscribed in a square. The diagonals of the square intersect at the center of both the square and the circle. The diagonals coincide with the diameter of the circle.

Square Circumscribed About A Circle

Illustration of a square, with diagonals drawn, circumscribed about a circle. This can also be described…

Illustration of a square, with 1 diagonals drawn, circumscribed about a circle. This can also be described as a circle inscribed in a square. The diagonal goes through the center of both the square and the circle and coincides with the diameter of the circle.

Square Circumscribed About A Circle

Illustration of a square, with 1 diagonals drawn, circumscribed about a circle. This can also be described…

Illustration of a square circumscribed about a circle. This can also be described as a circle inscribed in a square.

Square Circumscribed About A Circle

Illustration of a square circumscribed about a circle. This can also be described as a circle inscribed…

Illustration of an elongated square dipyramid that is formed by elongating a square bipyramid by inserting a square prism between the two congruent halves.

Elongated Square Dipyramid

Illustration of an elongated square dipyramid that is formed by elongating a square bipyramid by inserting…

Illustration of an elongated square dipyramid that is formed by elongating a square bipyramid by inserting a square prism between the two congruent halves. The elongated dipyramid pictured here is tilted, or rotated, approximately 45°.

Elongated Square Dipyramid

Illustration of an elongated square dipyramid that is formed by elongating a square bipyramid by inserting…

Illustration used to prove the Pythagorean Theorem, according to Euclid. A perpendicular is drawn from the top vertex of the right triangle extended through the bottom square, forming 2 rectangles. Each rectangle has the same area as one of the two legs. This proves that the sum of the squares of the legs is equal to the square of the hypotenuse (Pythagorean Theorem).

Euclid's Pythagorean Theorem Proof

Illustration used to prove the Pythagorean Theorem, according to Euclid. A perpendicular is drawn from…

Illustration that can be used to prove the Pythagorean Theorem, the sum of the squares of the legs is equal to the square of the hypotenuse.

Geometric Pythagorean Theorem Proof

Illustration that can be used to prove the Pythagorean Theorem, the sum of the squares of the legs is…

Illustration that can be used to prove the Pythagorean Theorem, the sum of the squares of the legs is equal to the square of the hypotenuse. The geometrical illustration depicts a 3,4,5 right triangle with the square units drawn to prove that the sum of the squares of the legs (9 + 16) equals the square of the hypotenuse.

Geometric Pythagorean Theorem Proof

Illustration that can be used to prove the Pythagorean Theorem, the sum of the squares of the legs is…

A visual illustration used to prove the Pythagorean Theorem by rearrangement. When the 4 identical triangles are removed, the areas are equal. Thus, proving the sum of the squares of the legs is equal to the square of the hypotenuse.

Pythagorean Theorem Proof by Rearrangement

A visual illustration used to prove the Pythagorean Theorem by rearrangement. When the 4 identical triangles…

Illustration of an 8-point star, created by two squares at 45° rotations, inscribed in a circle. This can also be described as a circle circumscribed about an 8-point star, or two squares.

Star Inscribed In A Circle

Illustration of an 8-point star, created by two squares at 45° rotations, inscribed in a circle.…

Leaves - simple; alternate; edge lobed (lobes entire). Outline - rounded. Apex - cut almost squarely across, with a shallow hollow, giving a square look to the upper half of the leaf. Base - usually heart-shape. Leaf - three to five inches long and wide; very smooth; with four to six lobes (two lobes at the summit; at the sides two, or two large and two small). Bark - of trunk, dark ash-color and slightly rough. Flowers - four to six inches across, greenish-yellow, marked within with orange, somewhat tulip-like, fragrant solitary. May, June. Found - from Southwestern Vermont to Michigan, southward and westward. Its finest growth is in the valley of the lower Wabash River and along the western slopes of the Alleghany Mountains. General Information - Among the largest and most valuable of the North American Trees. It is usually seventy to one hundred feet high, often much higher, with a straight, clear trunk, that divides rather abruptly at the summit into coarse and straggling branches. The wood is light and soft, straight grained, and easily worked, with the heart wood light yellow or brown, and the thin sap wood nearly white. It is very widely and variously used - for construction, for interior finish, for shingles, in boat-building, for the panels of carriages, especially in the making of wooden pumps and wooden ware of different kings. I asked a carpenter: "Hope, is n't it the tulip wood (which you call poplar*) that the carriage-makers use for their panels?" "Yes, and the reason is, because it shapes so easily. If you take a panel and wet one side, and hold the other side to a hot stove-pipe, the piece will just hub the pipe. It's the best wood there is for panelling." "Of all the trees of North America with deciduous leaves, the tulip tree, next to the buttonwood, attains the amplest dimensions, while the perfect straightness and uniform diameter of its trunk for upwards of forty feet, the more regular disposition of its branches, and the greater richness of its foliage, give it a decided superiority over the buttonwood and entitle it to be considered as one of the most magnificent vegetables of the temperate zone." - Michaux. *The name should be dropped. The tree is not a poplar. The tulip tree was very highly esteemed by the ancients; so much so that in some of their festivals they are said to have honored it by pouring over its roots libations of wine.

Genus Liriodendron, L. (Tulip Tree)

Leaves - simple; alternate; edge lobed (lobes entire). Outline - rounded. Apex - cut almost squarely…

Leaves - simple; indeterminate in position because of their smallness and closeness. They are arranged in four rows up and down the branchlets. In younger or rapidly growing sprouts the leaves are awl-shaped or needle-shaped, somewhat spreading from the branch, very sharp and stiff, placed in pairs (or sometimes in threes), usually about one fourth of an inch long, and with the fine branchlets, which they cover, rounded. In the older and slower-growing trees the leaves are scale-like and overlapping, egg-shape, closely pressed to the branchlets which they cover, and with the branchlets square. As the branchlets grow, the lower scales sometimes lengthen and become dry and chaffy and slightly spreading. Bark - brown and sometimes purplish-tinged, often shredding off with age and leaving the trunk smooth and polished. Berries - about the size of a small pea, closely placed along the branchlets, bluish, and covered with a whitish powder. Found - in Southern Canada, and distributed nearly throughout the United States - more widely than any other of the cone-bearing trees. General information - An evergreen tree, fifteen to thirty feet high (much larger at the South), usually pyramid-shaped, with a rounded base, but varying very greatly, especially near the coast, where it is often twisted and flattened into angular and weird forms. The wood is very valuable, light, straight-grained, durable, fragrant. It is largely used for posts, for cabinet-work, for interior finish, and almost exclusively in the making of lead pencils. The heart-wood is usually a dull red (whence the name), the sap-wood white. Among the most picturesque objects in the Turkish landscape, standing like sentinels, singly or in groups, and slender and upright as a Lombardy Poplar, are the black cypress trees (C. sempervirens). They mark the sites of graves, often of those which have long since disappeared. In America, more than any other northern tree, the red cedar gives the same sombre effect, whether growing wild or planted in cemeteries. The Common Juniper (J. communis, L.), common as a shrub, is occasionally found in tree form, low, with spreading or drooping branches, and with leaves resembling those of a young Red Cedar, awl-shaped and spreading, but arranged in threes instead of opposite.

Genus Juniperus, L. (Red Cedar)

Leaves - simple; indeterminate in position because of their smallness and closeness. They are arranged…

Leaves - compound (odd-feathered; leaflets, seven to nine); opposite; edge of leaflets slightly toothed or entire; entire at base. Outline - of leaflet, long oval or long egg-shape. Apex - taper-pointed. Base - somewhat pointed. Leaf/Stem - smooth. Leaflet/Stem - about one fourth of an inch long, or more; smooth. Leaf/Bud - rusty-colored and smooth. Leaflet - two to six inches long; pale beneath; downy when young, but becoming nearly smooth, except on the ribs. Bark - of the trunk, light gray. In very young trees it is nearly smooth, but it soon becomes deeply furrowed - the furrows crossing each other, and so breaking the bark into irregular, somewhat square or lozenge-shaped plates. Then in very old trees it becomes smooth again, from the scaling off of the plates. The branches are smooth and grayish-green. The young shoots have a polished, deep-green bark, marked with white lines or dots. Winged seeds - one and a half to two inches long, with the "wing" about one fourth of an inch wide, hanging in loose clusters from slender stems. The base of the seed it pointed and not winged. Found - in rich woods, from Southern Canada to Northern Florida and westward. It is most common in the Northern States. The finest specimens are seen in the bottom lands of the lower Ohio River basin. General Information - a tree forty to eighty feet high. Often the trunk rises forty feet without branching. Its tough and elastic timer is of very great value, being widely used in the manufacture of agricultural implements, for oars, and the shafts of carriages, and in cabinet-work. Fraxinus from a Greek word meaning "separation," because of the ease with which the wood of the Ash can be split. I find in the notes of an old copy of White's "Natural History of Selborne" this comment: "The Ash, I think, has been termed by Gilpin the Venus of British trees." Gerardes' "Herbal" comments: "The leaves of the Ash are of so great a vertue against serpents, as that the serpents dare not be so bolde as to touch the morning and evening shadowes of the tree, but shunneth them afarre off, as Pliny reporteth in his 16 book, 13 chap. He also affirmeth that the serpent being penned in with boughes laide rounde about, will sooner run into the fire, if any be there, than come neere to the boughes of the Ash." In Scandinavian mythology the great and sacred tree, Yggdrasil, the greatest and most sacred of all trees, which binds together heaven and earth and hell, is an Ash. Its roots spread over the whole earth. Its branches reach above the heavens. Underneath lies a serpent; above is an eagle; a squirrel runs up and down the trunk, trying to breed strife between them.

Genus Fraxinus, L. (Ash)

Leaves - compound (odd-feathered; leaflets, seven to nine); opposite; edge of leaflets slightly toothed…

A T-square is used to aid drawing straight and parallel lines. It consist of a thin straightedge, the blade, and a head fastened at right angles.

T-square

A T-square is used to aid drawing straight and parallel lines. It consist of a thin straightedge, the…

To test the straightness of the edge two T-squares may be placed together.

Testing The Edge of T-squares

To test the straightness of the edge two T-squares may be placed together.