E. Geometry and Measurement · Geometry and Measurement: Circles, Triangles, Measurement Systems, Pythagoras and Volume

Using the Pythagorean Relationship with Composite Shapes

E1.5 A1About 70 minutes

Learning goals

I can find hidden right triangles inside composite shapes, trapezoids, rhombuses and roofs.
I can use the Pythagorean theorem to find a missing side for a perimeter or a missing height for an area.
I can find the slant height or the height of a cone or a pyramid.
I can solve multi-step real-life problems with composite shapes.

Words to know

composite shape
A shape made of two or more simpler shapes, such as a rectangle with a triangle on top.

height (altitude)
The perpendicular distance from a base to the opposite vertex or side.

slant height
On a cone or pyramid, the distance from the apex down the sloping surface to the edge of the base.

isosceles trapezoid
A trapezoid whose two non-parallel sides are equal.

rhombus
A quadrilateral with four equal sides. Its diagonals cross at right angles and cut each other in half.

apex
The top point of a cone or pyramid.

A hook: how much siding?

A family is putting new siding and trim on the front of their house. The front is a rectangle with a triangular roof section on top. The plan shows the width, the wall height and the height of the roof peak, but not the length of the sloping roof edges. The trim goes around the outside, so they need those lengths.

Composite shapes often hide right triangles. Find them, and the Pythagorean theorem gives you every missing length.

Remember from Grade 8

You found the perimeter and area of composite shapes by splitting them into rectangles, triangles and circles. Now the missing pieces will often be slanted sides or heights, which you find with a right triangle.

Where right triangles hide

In an isosceles triangle or a roof, the height splits the base in half.
In a trapezoid, dropping heights from the shorter base makes right triangles at the ends.
In a rhombus or kite, the diagonals cross at 90°.
In a cone, the height, the radius and the slant height form a right triangle.
In a square pyramid, the height, half the base and the slant height form a right triangle.

The front of a house: a rectangle with an isosceles triangle on top. The roof height splits the 8 m base into two 4 m halves, making a 3-4-5 right triangle.

Worked example

Example 1: Siding and trim

The front of the house is a rectangle 8 m wide and 5 m tall, with a triangular roof whose peak is 3 m above the top of the walls. Find the length of trim around the outside and the area to be covered.

The height of the roof triangle splits the base into 4 m and 4 m.
Each sloping edge: √(4² + 3²) = √25 = 5 m.
Perimeter (outside only): 8 + 5 + 5 + 5 + 5 = 28 m. (The top of the rectangle is inside the shape, so it is not counted.)
Area: 8 × 5 + ½ × 8 × 3 = 40 + 12 = 52 m².

28 m of trim, 52 m² of siding (before subtracting doors and windows).

An isosceles trapezoid garden bed. The difference between the bases, shared equally by the two ends, makes a 6-8-10 right triangle at each end.

Worked example

Example 2: A trapezoid garden

An isosceles trapezoid garden has parallel sides 10 m and 22 m, and slanted sides of 10 m. Find its area and perimeter.

Each end overhang: (22 − 10) ÷ 2 = 6 m.
Height: h = √(10² − 6²) = √64 = 8 m.
Area: (10 + 22) ÷ 2 × 8 = 16 × 8 = 128 m².
Perimeter: 10 + 22 + 10 + 10 = 52 m.

A = 128 m², P = 52 m

Worked example

Example 3: A rhombus window

A rhombus-shaped window has diagonals of 30 cm and 40 cm. Find its side length, perimeter and area.

The diagonals cut each other in half at 90°, making 4 right triangles with legs 15 cm and 20 cm.
Side: √(15² + 20²) = √625 = 25 cm. Perimeter = 4 × 25 = 100 cm.
Area of a rhombus = ½ × d₁ × d₂ = ½ × 30 × 40 = 600 cm².

Side 25 cm, P = 100 cm, A = 600 cm²

Right triangles inside a cone and a square pyramid. The slant height is the hypotenuse; the height and the radius (or half the base) are the legs.

Worked example

Example 4: Cone and pyramid

(a) A cone has radius 5 cm and height 12 cm. Find its slant height. (b) A square pyramid has a base of 10 m and a height of 12 m. Find the slant height of a face and the length of a sloping edge.

(a) s = √(5² + 12²) = √169 = 13 cm.
(b) The slant height goes from the apex to the middle of a base edge: legs are 12 m and half the base, 5 m. Slant = √(25 + 144) = 13 m.
The sloping edge goes from the apex to a corner. It is the hypotenuse of a triangle with legs 13 m (the slant height) and 5 m (half a base edge): √(169 + 25) = √194 ≈ 13.93 m.

(a) 13 cm; (b) slant 13 m, edge ≈ 13.93 m

Worked example

Example 5: A lot with a slanted side

A building lot has a 30 m front, a 45 m back, and a depth of 20 m, with right angles at both ends of the 20 m side. Find the length of the fourth (slanted) side, the perimeter and the area.

The back is 45 − 30 = 15 m longer than the front. The slanted side is the hypotenuse of a right triangle with legs 15 m and 20 m.
Slanted side: √(225 + 400) = √625 = 25 m.
Perimeter: 30 + 20 + 45 + 25 = 120 m.
Area (trapezoid): (30 + 45) ÷ 2 × 20 = 750 m².

25 m; P = 120 m; A = 750 m²

Watch out!

Perimeter counts only the outside edges. Inside lines (like the top of the wall under a roof) do not count.
Halve the base of an isosceles triangle or pyramid before using Pythagoras.
A pyramid has two different sloping lengths: the slant height (to the middle of an edge) and the edge (to a corner). Read carefully.

Hands-on activity

Make a paper cone

You need: paper, a ruler, a pencil, a compass or a 10 cm string, tape, and scissors (ask a grown-up to help with cutting).

Draw a circle with a radius of 10 cm and cut it out. Cut it in half along a diameter.
Roll one half into a cone so the straight edges meet, and tape them. The slant height is the old radius: 10 cm.
Measure the radius of the cone's opening and the height of the cone (stand it on a table and measure straight up to the tip).
Check that r² + h² is close to 10² = 100.

Model answer: the curved edge of the half circle is π × 10 ≈ 31.4 cm, which becomes the cone's circumference, so r = 31.4 ÷ (2π) = 5 cm. Then h = √(100 − 25) ≈ 8.7 cm. Your measurements should be close.

Just for fun

What do you call a cone with its top cut off?

A frustum. (Not a "frustrated cone", although it might be.)

Your turn

An isosceles triangle has a base of 12 cm and equal sides of 10 cm. Find its height and area.

A shed front is a rectangle 6 m wide, 4 m tall, with a triangular roof whose peak is 4 m above the walls. Find the perimeter and area of the front.

An isosceles trapezoid has parallel sides 8 cm and 20 cm and slanted sides of 10 cm. Find its area.

A rhombus has diagonals 10 cm and 24 cm. Find its perimeter and area.

A cone has radius 9 cm and height 12 cm. Find its slant height.

A cone has slant height 17 cm and radius 8 cm. Find its height.

Find the height and area of an equilateral triangle with sides of 10 cm, to one decimal.

More practice: warm-up

What is the formula for the area of a triangle?

Legs 6 and 8. Hypotenuse?

What is the formula for the area of a trapezoid with parallel sides a and b and height h?

In a cone, which length is the hypotenuse: r, h or s?

An isosceles triangle has a base of 14 cm. How long is each half of the base?

More practice: on your own

An A-frame cabin's front is an isosceles triangle with a base of 8 m and sloping sides of 8.5 m. Find its height and area.

A kite has a 16 cm diagonal that is cut in half by the other diagonal, which is split into 6 cm and 15 cm parts. Find the kite's perimeter and area.

A square pyramid has a base of 16 m and a slant height of 17 m. Find its height.

A lot has a 40 m front, a 64 m back and a depth of 32 m, with right angles along the depth side. Find the slanted side, perimeter and area.

A shape is a 10 cm square with an isosceles triangle on top whose equal sides are 13 cm. Find its perimeter and area.

Challenge

Find the area of a regular hexagon with sides of 6 cm.

A square pyramid has a base of 10 cm and sloping edges of 13 cm. Find its slant height and its height.

A window is a rectangle 100 cm wide and 120 cm tall with a semicircle on top. A brace goes from a bottom corner to the very top of the arch. How long is it?

Check your understanding

Quiz

An isosceles triangle has a base of 16 cm and equal sides of 10 cm. What is its height?

A cone has r = 6 cm and h = 8 cm. Its slant height is:

A rhombus has diagonals 6 cm and 8 cm. Its side length is:

A house front is an 8 m wide rectangle with a roof triangle on top. Why is the top edge of the rectangle not in the perimeter?

An isosceles trapezoid has bases 6 m and 14 m and slanted sides 5 m. Its height is:

A square pyramid has a base of 12 m and a height of 8 m. Its slant height is:

For parents and teachers

Big idea: E1.5 includes problems that involve composite shapes. Students look for hidden right triangles to find slanted sides and heights, then use perimeter and area formulas. Cone and pyramid heights prepare for the volume lessons that follow.

Common difficulties:

Not halving the base of an isosceles triangle, roof, rhombus or pyramid.
Counting inside lines in a perimeter.
Mixing up slant height and height in cones and pyramids. Drawing the right triangle helps.

How to help: Make the paper cone together (help with scissors). Look at roofs, ramps and A-frames in your neighbourhood and ask your teen where the right triangle is.